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1  JK/^«^^--^-^K      //        (sC£&&£<pa~>&. 


Firct  Drift  of  a   Report 
en  the  FLVAC 

f  ';' 
John  von  !T 


Contract  No.  W-670-ORD-4926 
Between  the 
United  States  Army  Ordnance  Department 
and  the 
University  of  Pennsylvania 


V 

r 


re  ochool  of  Electrical  Engineering 
university  of  Pennsylvania 

June  30,  1945 


National  Bureau  of  Standards 

Division  12 

Data  Processing  Systems 


5jW     Smithsonian 
Institution 
Libraries 


Gift  of 
PAUL  CERUZZI 


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C" 


DNTENTS 
»  Sectio; 

? p(SS  1-°  3EFIII] 

00IX3B  i#1      Auton  u        Lgitalc     :  ,.    _ 

ScD(ftJ3       |_           Exact    :i   ;  :r  t  I  i           !    •  .      ions  of   such       systoni                                          i 

1.3       Die         "  i          ■■  i'     Ln  th>  r  m\   rica]            rial  pr    iucod  by  such 
a  s: 


;  .L       Ch   ck  Lng    ■     i    :errecti ng       dfund  (      n  r    I ,    au1 

Lbi         - 


UAIN  SUBDIVISIONS   M    I:  E  SYi 


5, 

i 

... 

5 

.3 

5, 

.4 

5. 

.: 

5 

.6 

5 

.7 

weed  for  subdi /isions  3 
First:      Central   ariU^    tic   part:      CA 

;ond:      Control   control    part:      CC  3 

.  ...        Third:      '.'■  rious                of  memory  r                :      (a)   -   i  it 

Third:      (Coat,.)     Memory:     M  6 

2,<  ,   CA   (tcgc.1      r:      C),   ..'   ar     together   Lh  jiati1       part. 

Af j  ind    jfforent  pari     :      [npul  put,  mediatin 

4  :            i1   id    .v.;  i,    Lh     i    aside.      Outsid      recording  medium:      It  o 

2 . 7       Fourth       Input :     I  7 

Fifth:      Dut]  ut:     0  7 

•  orison  of  M  and  R,  considering  (a)  -  (h)  in  2.4  7 

3.0  FROCEDURE  OF  DISCUSSION 

3.1   Program:   Discussion  of  all  divisions  (specific  parts) 

at  i  in  2,  together  with  the  jsscntial  decisions  9 

if  :•  "Zigzag"  iiscussion  of  the  specific  parts  9 

3.3   Automatic  checking  of  errors  10 


4.0  ELELENTS,  SYNCHRONISM  NEURON  ANALOGY 

4.1  Role  of  rel">y-like  elements.  Example.   Rcle  of  synchronism         LO 

4.2  Ne  iron;  .  synapses,  excitatory  and  inhil  Ltory  ty;  12 
.'...:   Desirability  of  using  vacuum  tubes  of  the  conventional 

radio  tub",  type 


5.0  PRINCIPLES  &VERNING  THE  A]      [C.       TIOM 

Vacuum  tub*  elements:  Gates  or  triggers  14 

Binary  vs.  I  icimal  system  15 

Duration  of  binary  multiplication  ••  16 

Tel'  •      perations  vs.  saving,  equipment  1' 

Role  of  very  high  speed  (vacuum  tubes):   Pri)      of 
successive  operation.  Time  estimal 
n  of  the  princif le 
F  irther  discussion  of  the  principle  20- 

Rational  Bureau  of  Standard* 
Division  12 
Data  Processing  Systems 


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C  0  N  T  E  N  T  S 
Section 

.  D  E-ELEME'ITS 

6.1  Re  isons  for  the  intr  lucti  r   •       thetii        ment  21 

6.2  1  . j  rip'  Lon  of  the  s '  \ .    :  -  - 

■  .    I  ynchr  dsm.  :ating  1     ;:.•..._:  r.k 

The  r  ..  .   E-el    •.  vii      Lpli  thresholds. 

1  1  '-ys  ?/, 

Com]       ;ith  vacuum  tul  . 

7.-'  CIRCUITS  FOfi  THE  ARITHMETICAL  OPERATI  WS  +,  X 

7-1       i!  of  fe  lin;»  ..  binary  i     s:   !  i  -.  •       iporal 

s  ucc  05  si 
7.2   E-  ler  at  netv,      id  bl  ck  syml  >  <ia 
7-3   The  adder 

J-L,        The    multiplier:   Memory  r  ;uirements  23 

Lscussion  of  the    lory.  D<  Lay,  ^9 

Discussion  of  delays  30 

7.7  plier:   Det  died  structure  31 

f.8        .   ■  .:_:  .  Lier:  Further  requirements  (timing,  local  input 

id  output).  33 


3.0  circuit:  for  the  arithmetical  operations  -,  -r, 

3.1  Treatment  of   the  sign  34 

8.2  The  subtracter  35 
.:'.3  The  divider:  Detailed  structure  36 
8.4   The  divider:  Further  requirements  (cf.  7.?)  38 


9.0  THE  BINARY  POINT 

9.1  The  main  roi^  of  the  binary  point:   For  X,  f  39 

7.2  Connection  with  the  necessity  of  omitting  15  ;its  ifter  x. 

Decision:   Only  nuiribers  between  -1  and  1 

9.3  Consequences  Ln  planning.  Rule:-:  for  th<   peration  +  ,  — ',  x  , -r     A.1 
/.4   Rounding  off:   Rule  and  E-element  network  <  42 

13.0       .  .  FOR  THE  ARITHME      PERATION-/.   OTHER  01 

1  .1   The  square  rootor:   Detailed  structure  43 

•  square  rooter:  Further  "observati 
o   List  •  of  operations:   +',  -',  X1,  f 
.  i,4   Exclusion  of  certain  further  operatic! 


*>  , 


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CONTENTS 
Section 

11.0  ORGANIZATION  OF' CA.   CQKPLETE  LIST  OF  OPERATIONS 

11.1  Input  and  output  of  CA,  connections  with  M  f>0 

11.2  The  operations  i,  j.  52 

11.3  The  operation  s  .,53 

11.4  Complete  list  of  operations:  +,  — ,  X,  -r,  sf ,     i,  j,.  s 

and  conversions  55 

12.0  CAPACITY  OF  THE  MEMORY  M.   GENERAL  PRINCIPLES 

12.1  The  cyclical  (or  delay)  memory  5b 

12.2  Memory  capacity:-  The  unit.  The  minor  cycle.  Numbers 

and  orders  57 

12.3  Memory  capacity:'  Requirements  of  the  types-  (a)  -  (h)  of  2.4  53 

12.4  Memory  capacity:  Total  requirements  64 

12.5  The  delay  memory:  Physical  possibilities      65 

12.i6   The  delay  memory:'  Capacity  of  each  individual  |  dl  |  and 

the  multiplication  time.-  The  number  of  ]  dl   ]'"s  needed  68 

12.7  Switching  vs.  temporal  succession  12 

12.8  The  iconoscope  memory  73 

13 .-0  ORGANIZATION  OF  M 


13«1  \    dl  /  and  its  terminal  organs  A  and  SG  79 

13.2  SG  and  its  connections  80 

13.3  The  two  on  states  of  SG  81 

13.4  SG  and  its  connections:  Detailed  structure  82 

13.5  The  switching  problem  for  the  SG  83 

14.-0  CC  AND  M 

14.1  CC  and  the  orders  84 

14.2  Remarks  concerning  the  orders  (b)  86 

14.3  Remarks  concerning  the  orders  (c)  86 

14.4  Remarks  concerning  the  orders  (b);  (Continued)  87 

14.5  Waiting  times.  Enumeration  of  minor  and  major  cycles.  88 

15.0  THE  CODE 

15.1  The  conterits  of  M  91 

15.2  Standard  numbers  91 

15.3  Orders  ,  92 
15-4  Pooling  orders  96 
15.5  Pooling  orders.  (Continued  .  97 
15. b  Formulation  of  the  code  98 


# 


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?  I  G  U  R  E  S 


Paee 


1.  Synchronization  -  clock  pulses  24 

2.  Threshold  2  nearon  by  combining  Threshold  1  neurons  25 

3.  Adder  27 
h.  Elementary  memory.   (E-element)  29 

5.  Line  of  E-elements                    "  29 

6.  Same  with  gate  network  30 

7.  Simple  valve  -  31 
3.  Simple  discriminator  .32 
9.  Multiplier  32 

10.  Complement  valve  36 

11.  .'  Subtractor                                        ,  36 

12.  •  '  Complete  discriminator  37 

13.  '  Divider  37 

14.  Rounding  off  valve  42 

15.  Square  rooter  44 

16.  Input  and  output  of  CA  50 

17.  Connections  of  input  and  output  in  CA          52 


18.  Amplification,  switching  and  gating  scheme  of  a  I  dl  69 

19.  Individual  and  serial  cycling'  of  a  j  dl  |  aggregate  (a) ,  (b)  69 

20.  Connections  of  a  j  dl  \   in  detail  80 

21.  SG',  preliminary  form  of  SG  82 

22.  Supplementary  connections  of  the  L  83 


£ 


.•.:         ■'~^X*0     Definitions'.'  .,■• 

1.1  The  considerations  which  follow  deal  with  the  structure 
of  a  very  high  speed  automatic  digital,  computing  system,  and  in  particu- 
lar with  its  logical  control.  Before  going  "into  specific  'details ,  so.-.e 
general  explanatory  remark*  regarding  these  concepts  may  be  appropriate. 

1.2  An  automatic  computing  system  is  a  (usually  highly  com- 
posite) device,  which  can  carry  out  instructions  to  perform  calculations 
of  a  considerable  order  of  complexity  -  e.g.  to  solve  a  non-linear  par- 
tial differential  equation  in'  2  or  3  independent  variables  numerically. 

The  instructions  which  govern  this  operation  must  be 

given  to  the  device  in  absolutely  exhaustive  detail.  They  include  all 
•  -•  ■  ••■■        ft  ..,'     ;       .  /,,•'   •:        ,.■  I-:. 
numerical  information  which  is  required  to  solve  the  problem  under  con- 
sideration: Initial  and  boundary  values  of  the  dependent  variables, 
values  of  fixed  parameters  (constants),,  tables  of  fixed  functions  which 
occur  in  the  statement  of  the  problem.  These  instructions  must  be  given, 
in  some  form  which  the  device  can  sense:  Funchc-d  into  a  system  of  punch- 
cards  or  on  teletype  tape,  magnetically  impressed  on  steel  tape  or  wire, 
photographically  impressed  on  motion  picture  film,  wired  into  one  or  more, 
fixed  or  exchangeable  plugboards  -  this  list  being  by  no  means  necessar- 
ily complete.  All  these  procedures  require  the  use  of  some  code,  to- 
express  the  logical  and  the  algebraical  definition  of  the  problem  under 
consideration,  as  well  as  the  necessary  numerical  material  (cf.  above). 

Once  these  instructions  are  given  to  the  device,  it  must 
be  able  to  carry  them  out  completely  and  without  any  need  for  further 
intelligent  human  intervention.  At  the  end  of  the  required  operations 
the  uevice  must  record  the  results  again  in  one  of  the  forms  referred  t*> 


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above.  The  results  are  numerical  data;  they  arc  a  specified  part  of  the 
numerical  material  produced  by  the  device  in  the  process  of  carrying  out 
the  instructions  Referred  to  above, 

1.3    It  is  worth  noting,  however,  that  the  device  will  in 
general  produce  essentially  more  numerical  irate  rial  (in  order  to  reach 
th  :  results)  than  the  (final)  results  mentioned.  Thus  only  a  fraction 
of  its  numerical  output  will  have  to  be  recorded  is  indicated  in  1.2,  the 

winder  will  only  circulate  in  the  int<  rior  of  the;  device,  and  never 
be  recorded  for  human  censing.  This  point  will  receive  closer  consider- 
ation subsequently,  in  particular  in 

\.U         The  remarks  of  1*2  on  the  desired  automatic  functioning 
of  the  device  must,  of  course,  assume  that  it  functions  faultlessly. 
Malfunctioning  of  any  device  has,  however,  always  a  finite  probability  - 
and  for  a  co-plicated  device  and  a  long  sequence  of  operations  it  may 
not  be  possible  to  Keep  this  probability  negligible.  Any  error  may 
vitiate  the  entir  output  of  the  device.  For  the  recognition  end  cor- 
rection of  such  malfunctions  intelligent  human  intervention  will  in 
general  be  necessary. 

However,  it  may  b  possible  to  avoid  even  these  pheno- 
mena to  some  extent.  The  device  may  recognize  the  mbst  frequent  mal- 
functions automatically,  indicate  their  presence  and  location  by  exter- 

ly  visible  signs,  and  then  stop.  Under  certain  conditions  it  eight 
even  carry  out  the  necessary  correction  automatical!;:     continue. 
(Cf.  .) 

_o_  ' 


!► 


V 


2.0  Main  subdivision  of  the  system  ■ 

2.1  In  analyzing  the  functioning  of  the  contemplated  device, 
certain  classificitory  distinctions  suggest  themselves  immediately. 

2.2  First:  Since  the  device  is  primarily  a  computor,  it 
will  have  to  perform  the  elementary  operations  of  arithmetics  most  f re- 

■ 

quontly.  There  are  addition,  subtraction,  multiplication  and  division": 
+■  ,  -,  x,-f  .  It  is  therefore  reasonable  that  it  should  contain  special- 
ized organs  for  just  these  operations. 

It  must  be  observed,  however,  that  while  this  principle 
as  such  is  probably  sound,  the  specific  way  in  which  it  is  realized 
requires  close  scrutiny.  Even  the  above  list  of  operations:  +,  -,  x,  -r  f 
is  not  beyond  doubt.   It  may  be  extended  to  include  such  operation  as../*", 

,  sgn,  1  1,  also  -^log,  log,  In,  sin  and  their  inverr.es,  etc.  One 
might  also  consider  restricting  it,  e.g.  omitting  -  and  even  x.  One 
might  also  considrr  more  clastic  arrangements.  For  some  operations  rad- 
ically different  procedures  are  conceivable,  e.g.  using  successive  ap- 
proximation methods  or  function  tables.  These  Matters  will  be  gone  into 
in  .  At  any  rate  a  central  arith- 

metical part  of  the  device  will  probably  have  to  exist,  and  this  consti- 
tutes the  first  specific  part:  CA. 

2.3  Second:  The  logical  control  of  the  device,  that  is  the  ■ 
proper  sequencing  of  its  operations  can  be  most  efficiently  carried  out 
by  i   central,  control  organ.   If  the  device  is  to  be  elastic,  that  is  a^ 
nearly  as  possible  '.11  purpose,  then  a  distinction  must  be  made  between 
the  specific  instructions  given  for  and  defining  a  particular  problem, 
and  the  generc-l  control  organs  which  see  to  it  that  these  instructions  - 

-3- 


no  matter  what  they  are^-  are  carried  out.  The  former*  must  be  stored 
in  some  way  -  in  existing  devices  this  is  done  as  indicated  in  1.2  - 
the  latter  are  represented  by  definite  operating  parts  of  the  device, 
3y  the  central  control  we  mean  this  latter  function  only,  and  the  organs 
which  perform  it  form  the  second  specific  part:  CC, 

2.1+        Third:  Any  device  which  is  to  carry  out  lonL;  and  c 
plicated  sequences  of  operations  (specifically  of  calculations)  must 
have  a  considerable  memory.  At  least  the  four  following  phases  of  its 
operation  require  a  memory: 

(a).  Even  in  the  process  of  carryiig  out  a  multiplica- 
tion  or  a  divisicn,  a  series  of  intermediate  (partial)  results  must  be 

remembered.  This  applies  to  a  lesser  extent  even  .to  additions  and  sub- 

i 
tractions  (when  a  carry  digit  nay  have  to  be  carried  over  several  posi- 

.s),  and  to  a  greater  extent  to  \j*tj   3/  ,   if  £nesg* operations  ire 

wanted,   (Cf.  .) 

(b)  The  instructions  which  Severn  a  Complicated  pr 

lem  may  constitute  -A   considerable  material,  particularly  so,  if  th>  code 

is  circumstantial  (which  it  is  in  most  arrangements).   This  material  must 

be  remembered. 
i 

(c)  in  many  problems  specific  functions  play  an  -  ssen- 

tiai  role.  They  are  usually  given  in  form  of  a  table.   Indeed  in  some 
ca^es  this  is  the  way  in  .'jhich  they  are  given  by  experience  (..'.g.  tJ 
equation  of  state  of  a  suostance  in  ;uany  hydre dynamical  problems),  in 
other  cases  they  may  be  given  by  analytical  expressions,  but  it  may 
nevertheless  be  simpler  .and  quicker  tc   bt  In  th  ir  values  from  a  f. 
tabulation,  than  to  compute  them  anew  (on  the  b  .sis  of  th...  analytical 


( 


definition)  whenever  as  value  is  required.   It  is  usually  convenient  to 
have  tables  of  a  moderate  number  of  entries  only  (100-200)  and  to  use 
interpolation.  Linear  and  even  quadratic  interpolation  -will  not  be 
sufficient  in  most  cases,  so  it  is-  best  to  count  on  a  standard  of  cubic 
cr  biquadratic  (or  .-von  higher  order)  interpolation,  cf.  . 

Some  of  the  functions  mentioned  in  the  course'  of  2.2 

10    2 
nay  be  handled  m  this  way:    lg,  rig,  In,  sin  and  their  inverses,  . 

possibly  also  N/  ,    .  ■  Even  the  reciprocal  might  be  treated  in  this 

manner,-  thereby  reducing  >f  -.to  x. 

(d)  For  partial  differential  equations  the  initial  con- 
ditions and  the  boundary  -Conditions-  may  ..constitute  an  extensive  numerical 
material,  -which  must,  be  remembered  throughout  :a  'given  problem.  . 

(e)  For  partial  differential. .equations  of  the  hyperbolic 
or  parabolic  type,  integrated  along  a  variable  t,  the  (intermediate)  re- 
sult's belonging  to  the  cycle  t  must  be  remembered  for  the  calculation  of 
the  cycle  t  +  dt.  -  This  material  is  much  of  the.  type  (d),  except  that  it 
is  not  put  into  the  devipe  by  human  operators,  but  produced  (and  probably 
subsequently  again  removed  and  replaced  by  the  corresponding  data  for 

t  ■*-  dt)  by  the  dev;c--  itself,  -in  the  course  of -its  automatic  operation.  . 

(f)  For  total  differential  equations  (d),  (e)  apply 
too,  but  they  r--q.ire  smaller  memory  capacities;  Further  memory  require-  - 
ments  of  the  type  (a)  are  required  in  problems  which  depend  on  given 
constants,  fixed  parameters,  ..etcr  . 

(g)  Problems  which  are  solved  by  successive  approxima- 
>  Ions  (e.g.  .partial  differential  equations  of  the  elliptic  type,  treated 
by  relaxation  methods ) ■,  require' a  memory "  of  the  tyj.  (e);   rhe  (intermedial 


-5- 


* 


£ 


£ 


results  of  each  approximation  must  be  ..remembered,  while  those  of  the  next 
-re  being  computed. 

(h)  Sorting  problems  and  certain  statistical  experi- 
ments (for  which  a  very  high  speed  device  offers  an  interesting  opportun- 
ity) require  a  memory  for  the  material  which  is  being  treated; 

2.5    To  sum  up  the  third  remark:  The  device  requires  a  con- 
siderable memory.  While  it  appeared,  that  various  part's  of  this  memory 
have  to  perform  functions  which  differ  somewhat  in  their  nature  and  con- 
siderably in  their  purpose,  it  is  nevertheless  tempting  to  treat  the 
entire  memory  as  one  organ,  and  to  have  its  parts  even 'as  interchangeable 
as  possible  for  the  various  functions  enumerated  above.-  This  point  will 
be  considered  in  detail  cf. 

At  any  rate  the  total  memory  -.constitutes  the  third-  specific 
oart  of  the  device:  M. 

*■  ■■■■— i i.--.  ■—  nam  »»■■■■■ 

2.6,    The  three  specific  parts  CA,  CC  together  C  and  M  corre-  ' 
spond  to  the  associative  neurons  in  the  human  nervous  system.  It  remains 
to  discuss  the  equivalents  of  the  sensory  or  afferent  and  the  motor  or 
effe-rqnt  n  jurons.  These  are  the  input  nnd  the  out  put  organs  of  the  de- 
vice,  and  we  shall  now  consider  them  briefly.  •• 

In  other  words:  All  transfers  of  numerical  (or  other) 
information  between  the  .parts  C  and  M  cf  the  device  must  be  effected  by 
the  mechanisms  contained  in  these  parts.  There  remains,  however,  the 
necessity  of  getting  the  original  definitory  infermation  from  utside 
int^  the  device,  and  also  cf  getting  the  final  information,  the  results, 
from  the  device  into,  the  outside.  ' 

By. the  outside  we  mean  media  cf  the  type  described  in 

-6- 


1.2:  Here  information  can  be  produced  more  or  less  directly  by  human 
action  (typing,  punching,  photographing  light  impulses' produced  bykeys 
of  the  same  type,  magnetizing  metal  tape  or  wire  in  some'  analogous  manner, 
etc.),  it  can  be   statically  stored,  and  finally  sensed  more  or  less  di- 
rectly by  human'  organs. 

The  device  must  be  endowed  with  the  ability  to  maintain 
the  input  and  output  (sensory  and  motor)  contact  with  some  specific  medium 
of  this  type  (cf.  1.2):  That  medium  will  be'  called  the  outside  recording 
medium  of  the  device:   R.   Mow  we  have: 

2.7  Fourth:  The  device  must  have  organs  to  transfer  (numer- 
ical or  other)  information  from  R  into  its  specific  parts  C  and  M.  These 
organs  form  its  input,  the  fourth  specific  part:  I.  It  will  be  seen, 
that  it  is  best  to  make  all  transfers' from -R  (by  I)   into  U,   and  never 
directly  into  C  (cf.  ). 

2.8  Fifth:  The  device  must  have  organs'  to  transfer  (pre- 
sumably only  numerical  information)  from  its  specific  parts  C  and  M  into 
R.  These  organs  form  its  output ,  the  fifth  specific  part:  0.   It  will 
'be  seen  that  it  is  again  best  to  make'  -all  transfers  from  M  (by  0)  into 
R,  and  never  directly  from  C  (cf.      *  ). 

2.9  The  output  information,  which  goes  into  R,  represents, 
of  course,  the  final  results  of  the  operation  of  the  device  on  the  prob- 
lem' under  consideration.  These  must  be  distinguished  from  the  intermed- 
iate results,  discussed. e.g.  in  2.4,  (e)-(g),  which'  remain  inside  M.  At 
this  point  .an  important  question  arises:  Quito  apart  from  its  attribute ■ 
of  more  or  less  direct  accessibility  to  human  action  and  perception  'ft 
has  also  the  properties  of  a  memory.  Indeed,  it  is  the  natural  medium 

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for  long  time  storage  of  all  the  information  obtained  by  the  automatic 
device  on  various  problems.  Why  is  it  then  necessary  to  provide  for  an- 
other type  of  memory, within  the  device  M?  Could  not  all,  or  at  least 
functions  of  M  -  preferably  those  which  involve  great  bulks  of  informa- 
tion -  be  taken  over  by  R? 

Inspection  of  the  typical  functions  of  >),,   as  enumerated 
in  2. h,    U)-(h),  shows  this:  It  would  be  convenient  to  s"hift  (a)  (the 
short-duration  memory  required  while  an.  arithmetical  operation  is  bein£ 
carried  out)  outside  the  device,  i.e,  from  id  into  R.   (Actually  (a)  will 

S    be  inside  the  device,  but  in  CA  rather  than  in  U.     Cf .  -th<  end  of  12,2) 
All  existing  devices,  even  the  existing  desk  computing  machines,  use  the 
equivalent  of  U   at  this  point.  However  (b)  (logical  instructions)  might 
be  sensed  from  outside,  i.e.  by  I  from  R,  and  the  same  goes  for  (c) 
(function  tables)  and  (e)t  (g)  (intermediate  results).  The  latter  may 
be  conveyed  by  0  to  R  when  the  device,  produces  them,  and  sensed  by  I 
from  R  when  it  needs  them.  The  same  is  true  to  some  extent  of  (d)  (ini- 
tial conditions7  and  parameters)  and  possibly  even  of  (f)  (intermediate 

«.    results  from  a  total  differential  equation).  Aj  to  (h)  (sorting  and 
statistics1),  the  situation  is  somewhat  ambiguous:   In  many  cat.es  the 
possibility  of  using  M'  accelerates  matters. decisively,  but  suitable 
blending  of  the  use  of  M  with  a  longer  range  use  of  R  may  be  feasible 
without  serious  loss  of  speed  and  increase  the  amount  of  material  that 
can  b..:  handled  considerably. 

•Indeed,  all  existing  (fully  or  partially  automatic) 
computing,  devicce  uso  R  -  as  a  .stack  of  punchcards  or  a  length  of 
teletype  tape  -  for  -all  these  ■ purposes  (excepting  (a),  as  pointed  out 

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above).  Nevertheless  it  will  appear  that  ">  really  high  speed  device 
would  be  very  limited  in  its  usefulness,  unless  it  can  rely  on  ..'., 
rather  than  on  R,  for  all  the  purposes  enumerated  in  2.L,    (a)-(h), 
with  certain  limitations  in  the  case  of  (c),  (g),  (h),  (cf. 
)• 

3.0  Froceduro  of  Discusgie:; 

3.1  The  classification  of  2.0  being  completed,  it  Ls  new 
possible  to  take  up  the  five  specific  parts  into  which  the  device  w  ls 
.seen  tc  be  Subdivided,  and  to  discuss  them  one  by  one.  Such  a  discussion 
must  oring  out  the  features  required  for  each  one  of  these  parts  in  it- 
self, as  well'  as  in  their  relations  to  each  ;thcr.   It  must  also  deter- 
mine the  specific  procedures  to  be  used  in  dealing  with  numbers  from  the 
point  of  view  of  the  device,  in  carrying  out  arithmetical  operations, 
and  providing  for  the  general  logical  control.   All  questions  of  timing 
and  of  speed,  and  of  the  relative  importance  of  various  factors,  must 

be  settled  within  the  framework  of  these  considerations. 

3.2  Tn-  ideal  procedure  would  be,  to  take  up  the  five  spe- 
cific parts  in  some  definite  order/ to  treat  each  one  of  them  exhaustive  Lj  , 
and  go  on  to  the  next  one  only  after  the  predecessor  is  completely  dis- 
posed of.  However,  this  seems  hardly  feasible.  Two  desirable  fe  itures 

of  the  various  parts,  and  the  decisions  based  .on  them,  emerge  only  after 
a  somewhat  zigzagging  discussion.   It  is  therefore  necessary  to  take 
up  one  part  first,  pass  after- an  incomplete  discussion  to  a  second  part, 
return  after  an  equally  incomplete : discussion  of  the  latter  with  the 

combined  results  t6  the  first 'part,' extend'- the  discussion  of  the  first 

i 

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part  without  yet  concluding  it,  then  possibly  go  on  to  a  third  part,  etc. 
Furthermore,  these  discussions  of  specific  parts  ..ill  be  nixed  with  dis- 
cussions of  general  principles,  of  arithmetical  procedures,  of  the  e 
ments  to  be  used,  etc. 

In  the  course  of  such  a  discussion  the  desired  features 
and  the  arrangements  which  seem  best  suited  tc  secure  then  will  ;rystaili2 
gradually  until  the  device  and  its  control  issume  a  fairly  i   finite 
As  emphasized  before,  this  applies  tc  the  physical  device  as  wexl  as  to 
the  arithmetical  and  logical  arrangements  which  govern  its  functioning. 

3.3    In  the  course  of  this  iiscussion  the  viewpoints  of  1./+, 
concerned  with  the  detection,  location,  and  under  certain  co)     >ns 
even  correction,  of  malfunctions  must  also  rec-jive  some  consideration. 
That  is,  attention  must  be  given  to  facilities  for  checkin-  errors.  We 
will  not  be  able  to  do  anything  like  full  justice  to  this  important  sub- 
ject, but  we  will  try  to  consider  it  at  least  cursorily  whenever  this 
seem;  ee5ential  (cf.  ). 

4.0  Eleminte,  Synchronism  Heuron  A.vilc;;y 

U.l         W";  begin  the  discussion  with  som-i   general  remarks: 

Every  digital  computing  device  contains  certain  relay 
like  elements,  with  discrete  jquilibria.   Such  nn  element  has  two  or   mcr 
distinct  states  in  which  it  can  exist  indefinitely.  These  may  b:  perfect 
equilibria,  in  each  of  which  the  element  will  remain  without  any  outsi 
support,  while  appropriate-  outside  stimuli  will  transfer  it  from  en- 
equilibrium  into  another.  Or,  alternatively,  there  may  be  two  states, 

a 

one  of  which  is  in  equilibrium  which  exists  when  there  is  no  outside 

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support,  while  the  other  depends  for  its  existence  upon  the  presence  of 
an  outside  stimulus.  The  relay  action  manifests  itself  in  the  emission 
of  stimuli  by  the  element  whenever  it  has  itself  received  a  stimulus --of 
the  typi  indicated  above,  The  emitted  stimuli  must  be  of  the  same  kind 
as  the  received'one,  that  is,  they  must  be  able  tc  stimulate  other  clc- 
ra  nts.   Fhere  nust,  however,  be  no  energy  relation  between  the  receive  i 

'■   the  emitted  stimuli,  that  is,  an  element  which  has  received  one 
stimulus,  must  be  able  to  emit  several  of  the  same  intensity.   In  other 
words:   Being  a  relay  the. element  must  receive  its  energy  supply  from 
another  sourcu  than  the  inc  u  i  ig  stimulus. 

In  existing  digital  cemputing  devices  various  mechan- 
ical, or  cloctric&l  devices  have  been  used  -as  elements:  Whoels,  which 
can  be  locked  into  any  one  of  ten  (or  more)  significant  positions,  and 
which  on  moving  fr  m  >ne  position  to  anoth  r  ti  msmit  electric  puis 
that  may  cause  other  similar  wheels  to  move;  single  or  combined  telegraph 
,    actuated  by  an  electromagnet  and  opening  or  closing  electric  cir- 
cuits';  cc   i  i  Lons  of  those  two  elements; — and  finally  there  exists  the 

i  tempting  possibility  of  using  vacuum  tubes,  the  grid  acting 
as  .  vali  i  for  the  cathode-plate  circuit.  In  the  last  mentioned  case 

;  •   j  alsc  be  replaced  by  deflecting  organs,  i.e.  the  vacuum  tube 
by  a  cathode  ray  tube — but  it  is  likely  that  for  some  time  to  come  the 

availability  and  various  electrical  advantages  of  the  vacuum 
tubes  proper  will  keep  the  first  procedure  in  the  foreground. 

Any  such  device  may  time  itself  autonomously,  by  the 
successive  reaction  times  of  its  elements.  In  this  case  all  stimuli 
i.  ultimately  originate  in  the  input.  Alternatively,  they  may  have 

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"their  timing  impressed  by  a  fixed  clock,  which  provides  certain  stimuli 
•that  are  necessary  for  its  functioning  pit  definite  periodically  recurrent 
moments'.  This  clock  may  be  a.  rotating  axis  in  a  mechanical  or  a  mixed, 
mechanico-electrical  device;-  and  it  may  be  an  electrical,  oscillator 
(possibly  crystal  controlled)  in  a  purely  electrical  device.   If  relianc: 
is  to  be  placed  on  synchronisms  of  several  distinct  sequences  of  opera- 
tions performed  simultaneously  bv  the  device-,  the  clock  impressed  timing 
is  obviously  preferable.  We  will  use  the  term  element  in  the  . above  de- 
fined technical  sense,  and  call  the  device  synchronous  or  asynchronous , 
according  to  whether  its  timing  is  impressed  by  a  clock  or  autono.o  ■  os , 
a  s  •  do  s  c  r  ib  ed  abo  ve. . 

4.2    It  is  worth  mentioning,  that  the  neurons  of  the  higher 
animals  -u    I  riitely  elements  in  the  above  sense..  They  have  all-or- 
ntene  character,  that  is  two  states:  Quiescent  o.nd  excited.  They  fulfill" 
the  requirements  cf  4.1  with  an  interesting  variant:  An  excited  neuron 
ts  the  standard  stimulus  along  many  lines  (axons).  Such  a  line  can, 
however,  be,  connected  in  two  different  ways  to  the  next  neuron:   First: 
In  an  .excitatory  syr.epsls,  so  that  the  stimulus  causes  the  excitation 
of  that  neuron.  S,cond:  In  an  inhibitory  synapsis,,  so  that  the  stimulus 
absolutely  prevents  the  excitation  of  that  neuron  by  *ny   stimulus  on  any 
other  (excitotory)  synopsis.  The  neuron  also  has  a  definite  reaction 
time,  between  thi  r.c  ption  of  a  stimulus  and  the  emission  of  the  stimuli 
caused  by  it,  tne  syro.ptic  delay. 

Following  W.  Fitts  and  W.  S.  [JacCulloch  ("A  logical 
calculus  of  the  ideas  immanent  in  nervous  activity",  Bull.  Math.  Bio- 
physics, Vol.  5  (1943),  pp  115-133)  ws  ignore  the  more  complicated  aspects 

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of  neuron  functioning:  Thresholds,  temporal  summation,  relative  inhibi- 
v>  ticn,  changes  of  the  threshold  by  after  effects  of  stimulation  beyond 

the  synaptic  delay,  .-tc.  It  is,  however,  convenient  to  consider  occasion- 

■ 

ally  noiiroris  with  fix:dd  thresholds  2  and  3»  that  is  neurons  which  can  be 
excited  Drily  by  (simultaneous)  stimuli  en  2  or  3  excitatory  synapses  (and 
none  on  an  inhibitory  synapsis),  Cf. 

It  is  c-  asily  seen,  that  thest  -simplified  neuron  functions 
can  be  imitated  b.\  telegraph  relays  or  by  vacuum  tubes.   Although  the  nerv- 
ous, system  is  presumably  isynchronous  (for  the  synaptic  iel  ys),,  precise 
synaptic  delays  can    Dbta-ined  by  using  synchronous  setups.  Cf. 

4-3    It  is  olear,  that  a  very  high  speed  computing  device 
should-  iiieally  have  vacuum  tube  elements.-..  Vacuum. tub  aggregat  s  Lik 

and  s  :alers  have  been  used  and  found  reliable  -at  reaction  times 
J     (synaptic  delays)  a.   l  rt  as  a  microsecond  (-  10   seconds),  this  is  a 
;  rf  ran  x  which  n    h  r  i   vice  can  approximate.  Indeed;'  purely 
mechanic  1  -device's  may  be  entirely  disregard  id  and  practical  telegraph 
relay  re-  3tion  times  irt   f  the  order  cf  10  milliseconds  (-  107-  seconds) 
sr  more.   It  Is  iat  r  sting  to  note  that  the  synaptic  time  of  a  human 
n  ur  i  U      I'   til   rd  r  of  a  milliseconds  (=  10""-'  seconds). 

In  th  considerations  vMhich  foil  w  we  will  assume  ac- 
•dinfily..  that  the  device  has  vacuum  tubes  as  elements.  We  will  also 

- 

try  tc  make  all  estimates  of  numbers  of  tubes  involved,  timing,  etc.  on 
the  basis,  that  the  types  of  tube's,  used,  are  the  conventional  and  com- 

-. -dally  available  ones.     That  is,   that  no  tubes  of  unusual  complexity 

■ 

.r  with  fundamentally  new  functions  are  tc  be  used.  The  possibilities 
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for  the  uso  of  new  types  of  tubes  will  actually  become  clearer  and  mere 
definite  after  a  th  r  ugh  analysis  with  the  conventional  types  (or  seme 
equiv  :.  ■■  demerits,  cf.  )  has  oeen  carried  out. 

Finf.lly  it  will  appear  that  a  synchronous  device  has 
censid;  -  /  1   i\     -  s    (cf .  ). 

5.0  Principles  r.vorning  the  'Arithmetic"!  Oper  tiuns 

5_-l    Let  us  now  consider  certain  functions  of  the  first  spe- 
cific part:  the  central  arithmetical  part  CA. 

;h-  element  in  the  sense  of  4.3,  ^he  vacuum  tube  used 
is  '.  current  v  iv  or  gate,  is  an  all-or-none  device,  or  at  least  it 

roxi  lite   :.    •      According  to  whether  the  rrid  bias  is  above  or  below 
cut-  >i'f;  i'  will  i'o:  currant  or  not.   It  ls  tru':  uhat  it  needs  definite 
:■:••    -         Ls  on  fti^  its  electrodes  in  order  to  maintain  either  state,  but 
there  are  combinations  of  vacuum  tubes  which  have  perfect  .equilibria: 
J  ■  "  i  states  in  each  of  which  the  combination  can  exist  indefinitely, 

3 

jiy  outside  support,  while  appropriate  outside  stimuli  (electric 
pulses)  will  transfer  it  from  one  equilibrium  into  another.  These  are 
the  so  called  trigger  circuits,  the  basic  one  having  two  equilibria  and 
co;.-  riod   or  one  pentode.  The  trigger  circuits  with  more 

than  two  equilibria  are  disproportionately  more  involved. 

Thus,  whether  the  tubes  are  used  as  gates  or  as  -triggers, 
the  ail-or-none,  two  equilibrium  arrangements  are  the  simplest  ones. 
Since  these  tube  arrangements  are  to  handle  numbers  by  means  of  their 
-  ,   digits,  it  is  natural  to-use  a  system  of  arithmetic  in  which  the  digits 
are  also  two  valued.   This  suggests  the  use  of  the  binary  system. 

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~V  The  analogs  of  human  neurons,  discussed  in  4.2  -  4.3 

are   equally  all-or-none  elements.   It  will  appear  that  they  are  quite 
useful  for  all  preliminary,  orienting  considerations  on  vacuum  .tube  sys- 
tems (cf .  ).   It  is  therefore  satisfactory 
that  here  too,  the  natural  arithmetical  system  to  handle  is  the  binary 
one. 

5.2    A  consistent  use  of  the  binary  system  is  also  likely 
to  simplify  the  operations  of  multiplication  and  division  considerably. 
Specifically  it  does  away  "with  the  decimal  multiplication  table,  or  with 
the  alternative  double  procedure  of  building  up  the  multiples  by  each 
multiplier  or  quotient  digit  by  additions  first,  and  then  combining 
these  (according  to  positional- value)  by  a  second  sequence  of  additions 
or  subtractions.  In  other  words:  .Binary  arithmetics  has  a<simpler  and 
more  one-piece  logical  structure  than  any  other,  particularly  than  the 
decimal  one. 

It  must  be  remembered,  of  course,  that  the  numerical 
material  which  is  directly  in.  human  use,  is  likely  to  have  to  be  ex- 
pressed  in  the  decimal  system.'  Hence,  the  notations  used  in  R  should 
be  decimal.   But  it  is  nevertheless  preferable  to  use  strictly  binary 
procedures  in  CA,  and  also  in  whatever  numerical  material  may  enter  into 
the  central  control  CC.  Hence  d   should  store  binary  material  only. 

This  necessitates  incorporating  decimal-binary  and 
binary-decimal  conversion  facilities  into  I  and  6.  Since  these  con- 
versions require  a' good  deal  of  arithmetical  manipulating,  it  is  most 
economical  to  use'  CA,  and  hence  for  coordinating  purposes  also  CC,  in 
conjunction  with  I'-andO.  The  use  of  CA  implies.^  however,  .that  all 

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-^  arithmetics  used  in  both  conversions  must  be-.' strictly  binary.  For 
details  cf.  . 

5.3    At  this  point  there  arises  another  question  of  principle. 
In  all  existing  devices  where  the  element  is  not  a  vacuum 
tube  the  reaction  time  of  the  element  is  sufficiently  long  to  make  a  .cer- 
tain telescoping  of  the  steps  involved  in  addition,  subtraction,  and  still 
more  in  multiplication  and  division,  desirable.  To  take  a  specific  case 
consider  binary  multiplication.  A  reasonable  precision  for  many  differ- 
encial equation  problems  is  given  by  carrying  3  significant  decimal  digits, 

-ft 
; ;  .that  is  by  keeping  the  relative  rounding-off  errors  below  10  .  This 

, -corresponds  to"  2"^  in  the.  binary  system  that  is  to  carrying  27  signif  i- 
■  cant -binary  digits.   Hence  a  •.multiplication  consists  of  pairing  each  'one 

of  27 .multiplicand  -digits  with  each  one  ti£   27  multiplier  digits,  and 
/)    forming  product  digits  0  arid  1  accordingly,  arid  then  positioning-  and  co.v.- 
bining  them.  These  are  essentially  27  'z   729  steps,  and  the  operations 
of  collecting  and  combining  may  about  double  their  number.  So  1000-1500 
steps  are  essentially  right. 
'  It  is  natural  to  observe  that  in  the  decimal  system  a 

considerably  smaller  number  of  steps  obtains:  8  -  6^  steps,  possibly 
doubled,  that  is  about  100  steps.  However,  this  low  number  is- pur- 
chased at  the  price  of  using  a  multiplication  table  or  otherwise  increas- 
ing or  complicating  the  equipment.  At  this  price  the  procedure  can  be 
shortened  by  more  direct  binary  artifices,  too,  which  will  be  considered 
presently.  For  this  reason  it  seems  not  necessary  to  discuss  the  deci- 
mal procedure  separately. 

.5.4   .  As  pointed  out  before,  100.0-f-1500  successive  steps  per 

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multiplication  would  make  any -non -Vacuum  tube  device  inacceptably  slow. 
All  such  devices,  excepting  some  of '•  the  latest  special  relays,  hive 
reaction  times  of  more  than  10  milliseconds,  and  these  newest  relays  (which 
may  have  reaction  times  down  to  5  milliseconds)  have  not  been  in  use  very 
long.  This  would  give  an  extreme  minimum  of  10-15  seconds  per  (8  decimal 
digit),  multiplication,  whereas  this  time  is  10  seconds  for  fast  modern  ' 
desk.:  computing  cachines,  and  6  seconds  for  the  standard  I«B.-M.  multipliers. 
(For  the  significance  of  these  djrations,  as  well  as  of  those  of  possible 
vacuum  tube  devices,  when  applied  to  typical  problems,  of.  ■  .) 

The  logical  procedure  to  avoid  these  long  durations, 
consists  of  telescoping  operations,  that  is  of  carrying  cut  simultaneously 
as  many  as  possible.  The  complexities  of  carrying  prevent  even  such  sim- 
ple operations  as  addition -or  subtraction  to  be  carried  out  at  once.  In 
division  the  calculation  of  a  digit  cannot  even  begin  unless  all  digits 
to  its  left  are  already  known.  Nevertheless  considerable  simultaneisa- 
tions  are  possible:  In  addition  or  subtraction  all  pairs  of  correspond- 
ing.digits  can  be  combined  at  once,  all  first  carry  digits  can  be  applied 
together  in  the  next  step,  etc.   In  multiplication  all  the  partial  pro- 
ducts cf  the  form  (multiplicand)  x  (multiplier  digit)  can  be  formed  and 
positioned  simultaneously — in  the  binary  system  such  a  partial  product. 
is  zero  or  the  multiplicand,  hence  this  is  only  a  matter  of  positioning. 
In  both  addition  and  multiplication  the  above  mentioned  accelerated  forms 
of  addition  and  subtraction  can  be  used.  Also,  in  multiplication  the 
partial  products  can  be  summed  up  quickly  by  adding  the  first  pair 
together  simultaneously  with  the  second  pair,  the  third  pair*  etc.;  then 
'adding;  the  first  pair  of  pair  sums  together  simultaneously  with  the 

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second  one,  the-third  one,  etc.;  and  so  on  until  all  terras  are  collected. 
(Since  27  4  2J ,   this  allows  to  collect  27  partial  sums  -  assuming  a  27 
binary  digit  multiplier  -  in  5  addition  times.  This  scheme  is  due  to 
H.  Aiken.) 

Such  accelerating,  telescoping  procedures  are  being 
used  in  all  existing  devices.   (The  use  of  the  decimal  system,  with  or 
without  further  telescoping  artifices  is  also  of  this  type,  as  pointed 
out  at  the  end  of  5.3.  It  is  actually  somewhat  less  efficient  than 
purely  diadic  procedures.  The  arguments  of  5.1  -  5.2  speak  against  con- 
sidering it  here.)  However,  they  save  time  only  at  exactly  the  rate  at 
which  they  multiply  the  necessary  equipment,  that  is  the  number  of  ele- 
ments in  the  device:   Clearly  if  a  duration  is  halved  by  systematically 
carrying  out  two  additions  at  once,  double  adding  equipment  w^.11  be 
required  (even  assuming  that  it  can  be  used  without  disproportionate 
control  facilities  and  fully  efficiently),  etc. 

This  way  of  gaining  time  by  increasing  equipment  i* 
fully  justified  in  non  vacuum  tube  element  devices,  where  gaining  time 
is  of  the  essence,  and  extensive  engineering  experience  is  available 
regarding  the  handling  of  involved  devices  containing  many  elements, 
A  really  all-purpose  automatic  digital  computing  system  constructed  along 
these  lines  must,  according  to  all  available  experience,  contain  over 
10,000  elements. 

5.5    For  a  vacuum  tube  element  device  on  the  other  hand,  it 
would  seem  that  the  opposite  procedure  holds  more-  promise. 

As  pointed  out  in  A.  3,  the  reaction  time  of  a  not  too 
complicated  vacuum  tube  device  can  be  made  as  short  as  one  microsecond. 

-18- 


f 


Now  at  this  rate  even  the  unmanipulated  duration  of  the  multiplication, 
obtained  in  5.3  is  acceptable:  1000-1500  reaction  times  amount  to  1-1.5 
milliseconds,  and  this  is  so  much  faster  than  any  conceivable  non  vacuum 
tube  device,  that  it  actually  produces  a  serious  problem  of  keeping  the 
device  balanced,  that  is  to  keep  the  necessarily  human  supervision  beyond 
its  input  and  output  ends  in  step  with  its  operations.   (For  details  of 
this  cf.  .)' 

Regarding  other  arithmetical  operations  this  can  be 
said:   Addition  and  subtraction  are  clearly  much  faster  than  multiplica- 
tion. On  a  basis  of  27  binary  digits  (cf.  5.3),  and  taking  carrying  into 
consideration,  each  should  take  at  most  twice  2.7  steps,  that  is  about 
30-50  steps  or  reaction  times.  This  amounts  to  .03-. 05  milliseconds. 
Division  takes,  in  this  scheme  where  shortcuts  and  telescoping  have 
not  been  attempted  in  multiplying  and  the  binary  system  is  being  used, 
about  the  same  number  of  steps  as  multiplication.   (cf. 

.)  Square  rooting  is  usually  and  in  this  scheme  too,  not, 
essentially  longer  than  dividing. 

5.6    Accelerating  these  arithmetical  operations  does  there- 
fore not  seem  necessary  -  at  least  not  until  we  have  become  thoroughly 
and  practically  familiar  with  the  use  of  very  high  speed  devices  of 
this  kind,  and  also  properly  understood  and  started  to  exploit  the 
entirely  new  possibilities  for  numerical  treatment  of  complicated  prob- 
lems which  they  open  up.  Furthermore  it  seems  questionable  whether  the 
method  cf  acceleration  by  telescoping  processes  at  the  price  of  multi- 
plying the  number  of  elements  required  would  in  this  situation  achieve 
'  its  purpose  at  all:  The  more  complicated  the  vacuum  tube  equipment — 

-19- 


r 


c 


that  is,  the  greater  the  number  of  elements  required — the  wider  the  tol- 
erances must  be.  Consequently  any-  increase  in  this  direction  will  also 
necessitate  working  with  longer  reaction  times  than  the  above  mentioned 
one  of  one  microsecond.  The  precise  quantitative  effects  of  this  factor 
are  hard  to  estimate  in  a  general  way — but  they  are  certainly  much  more 
important  for  vacuum  tube  elements  than  for  mechanical  or  for  telegraph 
relay  ones. 

Thus  it-  seems  worth  while  tc  consider  the  following 
viewpoint:  The  device  should  be  as  simple  as  possible,  that  is,  contain 
as  few  elements  as  possible.  This  can  be  achieved  by  never  performing 
two  operations  simultaneously,  if  this  would  cause  a  significant  increase 
in  the.  number  of-  elements  required.  The  result  will  be  that  the  device 
will  worK  more : reliably  and  the  vacuum. tubes  can  be  driven  to  shorter 
reaction  times  than  otherwise. 

3.7    The  point  to  which  the  application  of  this  principle 

can  be  profitably  pushed  will,  of  course,  depend  on  the  actual  physical 

- 
characteristics  of  the  available  vacuum  tube  elements.   It  may  be,  that 

the  optimum  is  not  at  a  10C#  application  of  this  principle  and  that  some 
compromise  will  be  found  to  be  optimal.  However,  this  will  always  de- 
pend on  the  momentary  state  of  the  vacuum  tube  technique,  clearly  the  • 
faster  the  tubes  are  which  will  function  reliably  in  this  situation,  the 
stronger  the  c?.se  is  for  uncompromising  application  of  this  principle. 
It  would  seem   that  already  with  the  present  technical  possibilities  the  , 
optimum  is  rather  nearly  at  this  uncompromising  solution. 

It  is  also  worth  emphasizing  that  up  to  now  all  think- 
ing about  high  speed  digital  computing  devices  has  tended  in  the  opposite 

-20- 


( 


t 


( 


direction:  Towards  acceleration  by  telescoping  processes  at  the  price 
of  multiplying  the  number  of  elements  required.   It  would  therefore  seem 
to  be  more  instructive  to  try  to  think  out  as  completely  as  possible 
the  opposite  viewpoint:  That  one  of  absolutely  refraining  from  the  pro- 
cedure mentioned  above,  that  is  of  carrying  out  consistently  the  principle 
formulated  in  5.6.  ( 

We  will  therefore  proceed  in  this  direction. 

6.0  E-elements 
6.1    The.  considerations  of  5.0  have  defined  the  main  princi- 
ples for  the  treatment  of  CA.  We  continue  now  on  this  basis,  with  some- 
what more. specific  and  technical  detail. 

In  order  to.  do  this  it  is  necessary  to  use  some  schematic 
picture  for  the  functioning  of.  the  standard  element  of  the  device:  In- 
deed, the  decisions  regarding,  the  arithmetical  and  the  logical  control 
procedures  of  the  device,' as  well  as  its  other  functions,  can  only  be 
made  on  the  basis  of  some  assumptions  about  the  functioning  of  the  ele- 
ments. 

The  ideal  procedure  would  be  to  treat  the  elements  as 
what  they  are  intended  to  be:  as  vacuum  tubes.  However,  this  would  n 
necessitate. a  detailed  analysis  of  specific  r^dio  engineering  questions 
at  this  early  stage  of  the  discussion,  when  too  many  alternatives  are 
still  open,  to  be  treated  all  exhaustively  and  in  detail.  Also,  the 
numerous  alternative,  .possibilities  for  arranging  arithmetical  proce- 
dures, logical  control,  etc.,  would  superpose  on  the  equally  numerous 
possibilities  for  the  choice  of  types  and  sizes  of  vacuum  t-ubes  and-  other 

circuit  elements  from  the  point  of  view  of  practical  performance,,  etc. 

All  this  would  produce  an  involved  and  opaque  situation  in  which  the 

'■'  ' 

-21- 


[r 


b 


preliminary  orientation  which  we  arc  now  attempting  would  be  hardly  possible. 

In  order  to  avoid  this  we  will  base  our  considerations 
on  a  hypothetical*  element,  which  functions  essentially  like  a  vacuum 
tube — e.g.  like  a  tritde  with  an  appropriate  associated  HLC-circuit — 
but  which  can  be  discussed  as  an  isolated  entity,  without  going  into 
detailed  radio  frequency  electromagnetic  considerations.  '..'e  re-empha- 
size: This  simplification  is  only  temporary,  only  a  transient  stand- 
point, to  make  the  present  preiLainary  discussion  possible.  After  the 

conclusion's  of  the  preliminary  discussion  the  elements  will  have  to  be 

i 

reconsidered  in  ttfeir  true  electromagnetic  nature;  ,  But  at  that  time  the 
decisions  of  the  preliminary  discussion  will  be  ■available,  and.  the 
corresponding  alternatives  accordingly  eliminated. 

6.2    '     The  analogs  of  "human  neurohs,  discussed  in  4.2-4.3  and 
again  referred  to  at  the  end  of  5.1,  seem  to-  provide  elements  of  just 
the  kind  postulated  at  the  end  of  6.1.  ■  V,'e  propose  to  use  them  accord- 
ingly for  the  purpose  described  there:  as  the  constituent  elements  of 
the  device,  for  the  duration  of  the  preliminary  discussion.  We  must 
therefore  give  a  precise  account  of  the  properties  which  we  postulate 
for  these  elements. 

Th<  element  which  we  will  discuss,  to  be  called  an 
£--:  lei.-ient,  will  be  represented  to  be  a  circle  0,  which  receives  the 
excitatory  and  inhibitory  "stimuli,  and  emits  its  own  stimuli  along  a  line 
attached  to  it:   0—.  This  axis  may  branch:  '0— <-,  0-^-.  The  emission 
along  it  follows  the  original  stimulation  by  a  synaptic  delay,  which  we 
can  assume  to  be  a  fixed  time,  the  same  for  all  E-elements,  to  be  denoted 
by  t.  We  propose  to  neglect' the  other -delays ■ (due  to-, conduction  of  the 


w 


c 


stimuli  along  the  lines)  aside  of  t.  We  will  mark  the  presence  of  the 
delay  t  by  an  arrow  on  the  line:  0-y— ,  0  >•■  -C  This  will  also  serve 
to  identify  the  origin  and  the  direction  mi  the  line. 

6.3    At  this  point  the  following  observation  is  necessary.  . 
In  the  human  nervous  system  the  conduction  times  along  the  lines  (axons) 
can  be  longer  than  the  synaptic  delays,  hence  our  above  procedure  of 
neglecting  them  aside  of  t  would  be  unsound.  In  the  actually  intended 
vacuum  tube  interpretation,  however,  this  procedure  is  justified:  t  is 
to  be  about  a  microsecond,  an  electromagnetic  impulse  travels  in  this 
time  300  meters,  and  as  the  lines  are  likely  to  be  short  compared  to  this, 
the  conduction  times  may  indeed  be  neglected.   (It  would  take  an  ultra 
high  frequency  device t  «  10~°  seconds  or  less-x-to  vitiate  this  argu- 
ment. ) 

Another  point  of  essential  divergence  between  the 
human  nervous  system  and  our  intended  application  consists  in  our  use 
of  a  well  defined  dispersionlcss  synaptic  delay  t,  common  to  all  E-ele- 
ments.   (The  emphasis  is  on  the  exclusion  of  a  dispersion.  We  will 
actually  use  E-elements  with  a  synaptic  delay  2t,  cf.  .) 

•We  propose  to  use  the  delays  t  as  absolute  units  of  time  which  can  be 
relied  upon  to  synchronize  the  functions  of  various  parts  of  the  device. 
The  advantages  of  such  an  arrangement  are  immediately  plausible,  specific 
technical  reasons  will  appear  in 

In  order  to  achieve  this,  it  is  necessary  to  conceive 
the  device  as  synchronous  in  the  sense  of  i*.l.  The  central  clock  is 
best  thought  of  as  anElectrical  oscillator,  which  emits  in  every  period 
t  a  short,  standard  pulse  of  a  length  t'  of  about  l/5t  -  l/2t.  The 

-23- 


€ 


€ 


<L 


1 


FIGURE       1 

>    i' 

_ 

<        > 

i — 

>     i 

r 

C                 > 

-  -i 

i  : 
i 

<• 

<v 

1-  - 

t 

i 

"LOCH       PUl"iii. 

1 
i 

! 

FoR       Ti-IEL     O  P  £  N       ! 

stimuli  emitted  nominally  by  an  E-elemont  are  actually  pulses  of  the 
clock,,  for  which  the  pulse  acts  as  a  gate.  There  is  clearly  a  wide 
tolerance  for  the  period  during  which 
the  gate  must  be-  kept  open,  to  pass 
the  clock-pulse  without  distortion. 
Cf.  Figure  1.  Thus  the  opening  of 
the  gate  can  be  controlled  by  any 
electric  delay  device  with  a  mean 
delay  time  t,  but  considerable  per- 
missible dispersion.  Nevertheless 
the  effective  synaptic  delay  will  be 
t  with  the  full  precision  of  the 
clock y  and  the  stimulus  is  completely 
renewed  and  synchronized  after  each  step.  For  a  more  detailed  descrip- 
tion in  terms  of  vacuum  tubes,  cf.. 

6./+    Let  us  now  return  to  the  description  of  the  E-element s. 

An  E-element  receives  the  stimuli  of  its  antecedents 

ss  excitatory  synapses:'  0— >-,  or  inhibitory  synapses:   0— >-. 

As  pointed  out  in  /+.2,  we  will  consider  E-element s  with  thresholds  1,  2, 
3,  that  is,  which  get  excited  by  thesr  minimum  numbers  of  simultaneous 
excitatory  stimuli.   All  inhibitory  stimuli,  on  the  other  hand,  will  be 
assumed  to  be  absolute.  E-elements  with  the  above  thresholds  will  be 
denoted  by  0,[2j,  \3J,   respectively. 

Since  we  have  a  strict  synchronism  of  stimuli  arriving 
only  at  times  which  are  integer  multiples  of  t,  we  may  disregard  pheno- 
mena cf  tiriruj,  facilitation,  etc,  We  also  disregard  relative  inhibition. 


-21+- 


f 


( 


temporal  summation  Of  stimuli,  changes  of  threshold,  changes  synapses, 
etc.  In  all  this  we  are  following  the  procedure  of  W.  Fitts  and  W.  J.. 
MacCulloch  (cf.  Loc.  cit.  k»Z).     We  will  also  use  E-elements  with  double 
synaptic  delay  2tj  Q-W,  and  mixed  types:  0-d-«^>-  _ 

The  reason  for  our  using  these  variants  is,  that  they 
give  a  greater  flexibility  in  putting  together  simple  structures,  and 
they  can  all  be  realized  by  vacuum  tube  circuits  of  the  same  complexity. 

It  should  be  observed,  that  the  authors  quoted  above 
-%)   have  shown,  that  most  of  these  elements  can  be  built  up  from  each  other. 
Thus  0->->-  is  clearly  equivalent  to  0-v-0->-,  and  in  the  case  of  [2  }  -»-  at 
least  —  (2j  ->->-  is  equivalent  to  the 
network. of  Figure  2,  However,  it 
-i  would  seem  to  be  misleading  in  our 
application,  to  represent  these 
functions  as  if  they  required  2  or  3  E-eleraents,  since  their  complexity 
in  a  vacuum  tube  realization  is  not  essentially  greater  than  that  of  the 
simplest  E—  element  0->-,  cf. 

We  conclude  by  observing  that  in  planning  networks  of 
E-elements,  all  backtracks  of  stimuli  along  the  connecting  lines  must 
be  avoided.  Specifically:  The  excitatory  and  the  inhibitory  synapses 

and  the  emission  points that  is  the  three  connections  on  ~^  )— > 

'  will  be  treated  as  one-way  valves  for  stimuli from  left  to  right  in  the 

above  picture.   But  everywhere  else  the  lines  and  their  connections ^r^-v 
will  be  assumed  to  pass  stimuli  in  all  directions.  For  the  delays  — ^ — 
either  assumption  can  be  made,  this  last  point  does  not  happen  to  matter 
Jj    in  our  networks. 

-25- 


* 


'C 


6. 5    Comparison  of  some  typical  E-element  networks  with  their 
^vacuum  tube  realizations  indicates,  that  it  takes  usually  1-2  vacuum  tubes 
for  each  E-element.   In  complicated  networks,  with  many  stimulating  lines 
for  each  E-element,  this  number  may  become  somewhat  higher.  On  the 
average,  however,  counting  2  vacuum  tubes  per  E-element  would  seem  to 
be  a  reasonable  estimate.  This  should  take,  care  of  amplification  and 
pulse-shaping  requirements  too,  but  of  course  not  of  the  power  supply. 
For  seme  of  the  details,  cf. 

7.0  Circuits  for  the  arithmetical  operations  4-  ,  X 

7.1    For  the  device and  in  particular  for  CA a  real  num- 
ber is  a  sequence  of  binary  digits.  We  saw  in  5.3,  that  a  standard  of  27 
binary  digit  numbers  corresponds  to  the  convention  of  carrying  8  signi- 
^  ficant  decimal  digits,  and  is  therefore  satisfactory  for  many  problems. 
We  are  not  yet  prepared  to-  make  a  decision  on  this  point  (cf .  however, 

),  but  we  will  assume  for  the  time  being,  that  the 
standard  number  has  about  30  digits. 
*J  |  When  an  arithmetical  operation  is  to  be  performed  on  such 

numbers,  they  must  be  present  in  some  form  in  the  device,  and  more  partic- 
ularly in  CA.  Each  (binary)  digit  is  obviously  representable  by  a  stim- 
ulus at  a  certain  point  and  time  in  the  device,  or  more  precisely,  the 
value  1  for  that  digit  can  be  represented  by  the  presence  and  the  value 
0  by  the  absence  of  that  stimulus.  Now  the  question  arises,  how  the 
30  (binary)  digits  of  a  real  number  are  to  be  represented  together.  They 
could  be  represented  simultaneously  by  30  (possible)  stimuli  at  30  differ- 
ent positions  in  CA,  or  all  30  digits  of  one  number  could  be  represented 

-26- 


^ 


© 


by  (possible)  stimuli  at  th  same  point,  occurrin  success 

periods  T  in  time. 

Following  the  principle  of   5.6 to  place  multiple  evr ■■ 

in  temporal  succession  rather  than  in  (simultaneous)  spacial  juxtapositi 

we  choose  the  latter  alternative.  Hence  a  number  is  represenl  -  I  by  a 

,  which  e.rl1  s  during  %   successive  periods  "f  the  stimuli  correspond-. 
I   ;    inary)  digits. 

7.2  In  the  following  discussions  we  will  draw  various  net- 
works of  S-elements,  to  perform  various  functions.  These  drawings  will 
also  be  used  to  define  block  symbols.  That  is,  after  exhibiting  the 

ucture  of  a  particular  network,  a  block  symbol  will  be  assigned  to  it, 

.".-hich  will  represent  it  in  all  its  further  applications including  those 

where  it  enters  as  a  constituent  Into  a  higher  order  network  and  its 
bol.   A  bi;ck  symbol  shows  all  input  and  output  lines  of  it; 
..  twork,  but  not  their  internal  connections.  The  input  lines  will  I 

■■■:  !  ~> ,  and  the  output  lines  — *   .   A  block  symbol  carries  the 

abbreviated  name  of  its  network  (or  its  function),  and  the  number  of 
Lements  in  it  as  an  index  to  the  name.  Cf .  e.g.  Figure  3.  below. 

7.3  v;e  proceed  to  describe  an  adder  n<  tworks  Figure  3. 
no   addends  come  in  on  the  input  lines- a1-,  ..a",  and  the  sum- is 

■  i  with  a  delay  2  ~~L 


tht  addend  inputs  on  the 
_in?  s.   (The  dotted  sxtra 
it  line  c  is  for  a  special  pur- 
pose which  will  appear  in  3.2)  The 
carry  •  i  by  '2   .   The 


.FlGL(-R~L    3 


4g>':.'6 


>- 


if 


1 


-27- 


• 


a 


^  corresponding  digits  of  the  two  addends  together  with  the  proceeding 

carry  digit  (delay  jl)   excite  each  one  of  0  (left),  (£),    (T),  and  an  out- 
put stimulus  (that  is  a  sura  digit  1)  results  only  when  0  is  excited  with- 
out (2\ ,  or  when  (y\    is  excited — that  is  when  the  number  of  l's  among  the 
three  digits  mentioned  is  odd.  The  carry  stimulus  (that  is  a  carry  digit  i) 
results,  as  pointed  out  above,  only  when  (2)  is  excited — that  is  when  there 
are  at  least  two  l's  among  the  three  digits  mentioned.  All  this  consti- 
tutes clearly  a  correct  procedure  of  binary  addition. 

In  the  above  we  have  made  no  provisions  for  handling  the 
si^n  of  a  number,  nor  for  the  positioning  of  its  binary  point  (the  analog 
of  the  decimal  point).  The^e  concepts  will  be  taken  up  in  ., 

but  before  consid«ring  them  we  will  carry  out  a  preliminary  discussion 
of  the  multiplier  and  the  divider,. 

7 .k         A  multiplier  network  differs  qualitatively  from  the  adder 
in  this  respect:   In  addition  every  digit  of  each  addend  is  used  only  once^, 
in  multiplication  each  digit  of  the  multiplicand  is  used  as  many  times  as 
there  are  digits  in  the  multiplier.  Hence  the  principle  of  5-6..   (cf . 
W)     also  the  end  cf  7-1)  requires,  that  both  factors  be  remembered  by  the 
.multiplier  network  for  a  (relatively)  considerable  time:  Since  each 
number  has  30  digits,  the  duration  of  the  multiplication  requires  remem- 
bering for  at  least  yp-  .-   900  periods  X   •   In  other  words:   It  is  no 
longer  possible,  as  in  the  adder,  to  feed  in-  the  two  factors  on  :.two  in- 
put lines,  and  to  extract  in  continuous  operation  the  product  on  the 
output  line — the  -multiplier  needs  a. memory  (cf.  2.4,  (a)). 

•  In  discussing  this  memory  we  need  not  bring  in  M — this 
is  a  relatively  small  memory  capacity  required  for  immediate  use  in  CA, 

v  and  it  is  best  considered  in  CA. 

-28- 


7«5    Tne  E-elements  can  be  used  as  memory  devices:   An  element 
which  stimulates  itself,  \^)ZZZ^ >  will  hold  a  stimulus  indefinitely. 
Provided  with  two  input  lines  rs,  cs  for  receiving  and  for  clearing 
(forgetting)  this  stimulus,  and  with  an  output  line  os  to  signalize  the 
presence  of  the  stimulus  (during  the  time  interval  over  which  it  is 
remembered),  it  becomes  the 


^ 


network  of  Figure  4.  ' 

It  should 
be  noted  that  this   m^ 


ft  C,uK£     * 


rs 
c  s 


Cfc± 


OS      t= 


-X 

y~\ 

i 

corresponds  to  the  actual  vacuum  tube  trigger  circuits  mentioned  at  the 
beginning  of  5.1.   It  is  worth  mentioning  that 


•1 


contains  one  E-ele- 


3 


ment,  while  the  simplest  trigger  circuits  contain  one  or  two  vacuum' tubes 
(cf.  loc.  cit.),  in  agreement  with  the  estimates  of  6.5. 
Another  observation  is  that 


m. 


remembers  only  one 


stimulus,   that  is.  one  binary  digit.      If  k-fold  memory  capacity  is  wanted, 


then  k  blocks 
ments 


mi 


* 


are  required,  or  a  cyclical  arrangement  of  k  E-ele- 
:  r- — C~\~}>---(~~\iF-  -  --y(~y^-\   •''•'Tflfi-s  cycle  can  be  provided  with 
inputs  and  outputs  in  various  ways ,  which  can  be  arranged  so  that. when- 
ever a  new  stimulus  (or  rather  the  fact  of  its  presence  or  absence,  that 
is  a  binary  digit)  is  received  for  remembering — say  at  the  left  end  of 
the  cycle — the  old  stimulus  which  should  take  its'-  place — coming  from  the 
right  end  of  the  cycle — is  automatically'  cleared.  Instead  of  going  into 
these  details,  however,  we  prefer  to  keep  the  cycle  open:-^\-^-(^y ...  a/^Vv 
and  provide  it  'with  such  terminal 


equipment  (at  both  ends,  possibly 


' 


ft  Cum    ^ 


■'<■  .       • — - — r 


""ft.    £  -  fc  l  e '  m  en"^  S 


connecting  them)  as  may  be  rciuirec 
in  each  particular  case.  This  simple  line  is  shown  again  in  Figure  5. 

-29- 


( 


I 


' 


Terminal  equipment,  which  will  normally  cycle  "the  output  os.  at  '  Ik   j's 
right  and  back  into  the  input  at  its  left  end,  but"  upon  stimulation  at 
s  suppress  (clear)  this  returning  of  the.,  output  os  and  connect  instead. 
the  input  with  the  line  rs, 


is  shown  in  Figure  6 

1A 


Ik. 


with  the  terminal  equipment 
of  Figure  6.,  is  a  perfect 


-nc,  urn     c 


rs 


<3S 


memory  organ,  but  without  it,  in  the  form  of  Figure  5.,  it  is  simply  a 
delay  organ.   Indeed,  its  sole  function  is  to  retain  any  stimulus  fcr 
k  periods  t  and  then  reemit  i\    and  to  be  able  to  do  this,  for  successive 
stimuli  without  any  interference  between  them. 

This  being  so,  and  remembering  that  each  E-elenent 
represents  (one  or  two)  vacuum  tubes,  it  would  seem  wasteful  to  use  k  - 
2k  vacuum  tubes  to  achieve  nothing  more  than  a  delay  kt.  There  exist 
delay  devices  which  can  do  this  (in  our  present  situation  t  is  about  a 
microsecond  and  k  is  about  30}  "more  simply. '  We  do  not  discuss  them  here, 
but  merely  observe  that,  there  arc  several  possible' arrangements   (cf. 
12.5).  Accordingly,  we  replace  the  block 
block 


Ik 


of  Figure  5  by  a  new 


dl  (k) 


,  which  is  to  represent  such. a  device.   It  contains 
no  E-elemont,  and  will  itself  be!  treated  as  a  new  element. 
We  observe,  that  is  '  dl  (k) 


is  a  linear  delay  cir- 


cuit, stimuli  can  backtrack  through  it  (cf.  the  end  of  6.4)«  To  prevent 
this,  it  suffices  to  protect  its  ends  by  E -elements,  that  is  to  achieve 

the  first  and  the  last  t  delay'  by  — (T)^ or  to  use  **  in  some 

combination  like  Figure  6,  where  the  E-elements  of  the  associated  network 

-30- 


r 


r 


I 


provide  this  protection. 

7.7    We  can  now  descrloe  a  /multiplier  network. 

Binary  multiplication  consists  of  this:   For  each  digital 
position  in  the  multiplier  (going  from  left  to  right),  the  multiplicand 
is  ;hifte  i  by  one  position  to  the  right,  j.;ih  then  it  is  or  is  not  added 
to  the  sum  of  parti  a  products  already  formed,  according  to  whether  the 
multiplier  digit  under  consideration  is  1  jr  0. 

Consequently  the  multiplier  must  contain  an  auxiliary 
network,  which  will  or  will  not  pass  the  multiplicand  into  the  adder, 
a>?_rainr;  to  whether  the  multiplier  iir.it  in  -uestion  is  1  or  0.  This 
can  be  achieved  in  two  steps:   Fir.?+,  a  network  is  required,  which  will 
emit  stimuli  during  a  certain  interval  of    7  periods  (the  interval  in 
which  the  multiplicand  is  .vanted),  provided  that  a  certain  input  (con- 
nected to  the  organ  which  contains  the  aultiplier)  was  stimulated  at  a 
cert  air:  earlier  moment  (when  the  proper  multiplier  digit  is  emitted). 
Such  a  network  will  be  called  a  ai^cri.minator.  Second,  a  valve  is  re- 
hired which  will  pacs  i   stimulus  only  if  it  is  also  stimulate:!  on  a 
second  input  it  possesses.   Ihese  two  clocks  together  solve  our  problem: 
The  discriminator  mast  be  properly  controlled,  its  output  connected  to 
the  second  input  of  the  valve,  and  the  multiplicand  routed  through  the 
valve  into  the  adder.  Th  : 
valve  is  juite  simpL  : 
Figure  7.  The  main  stim- 
ulus is  passed  :  r<  ...  is  to 
os,  the  second  input  centers  at  s. 


-31- 


'  I 


i 


9 


~) 


) 


) 


A .discriminator 
is  shown  on  Figure  8.  A  stimulus 
at  the  input  t  defines  the  mo-: 
ment  at  .-Jhich  the  stimulus,  which 
determines  whether  the  lat^r 
emission  (at  os)   shall  take  place 


flQi/RE       6 


Q^ 


at  all,  must  be  r;-cr;iv.ej  at  the  inputs.   If  these  two  stimuli  coincide, 
the  left  \2)   is  excited.   Consid<  ri    Its  feedback,  it.  will  remain  excited 
until  it  secceeds  in  stimulating  the  middle  {2)   .".'  The  middle  (k)   is  con- 
nected to  (is)  in  such  a  .-manner  that  it  can  be  excited  by  the  left  (2\ 
only  at.  a  moment  at  which  (is)  is  stimulated,  but  at  whose  predecessor 

(is)  was  not  stimulated — that  is  at  the  beginning  of  a  sequence  of 
stimuli  at  (is)  .   The  mi  idle  (£)   then,  ]uenches  the  left  (J2)    ,  and  to- 

■  t.her  with  (is)  excites  the  right  [2)    .   The  ~lidle  {2  )   now  becomes  and 


stays  quiescent  until  the  end  of  this  secuence  of  stimuli  at  n.s)    and 
beyond  this,  until  the  beginning  of  the  next  sequence.  Hence  the  left 

(2)  is  isolated  from  "the  two:  other  (2)  ,  and  thereby  is  ready  to  register 
the  s,  t  stimuli  for  the  next  (is)  sequence.  On  the  other  hand  the  feed- 
back of  the  right  (?)  is  such,  that  it  will  stay  excited  for  the  luratic 
of  this  (is;  sequence,  and .emit  stimuli'  at  cs.  There  is  clearly  a  delay 
2t  between  the  input  at  (is)  and  the  output  at  os. 
Now  the 


.on 


multiplier  network  ■cah.be 
put   together:      Figury.9. 
.The  multiplicand   circulates 
through 


dl  I 


t  he- 


multiplier  through 


al  II 


F  1  CWRZ      9 


a 


-\     cd    1 


.V. 


L_ 


<M   H 


1 — y—M.-) 


at. 1  m 


h 


-32- 


<4 


and  the  sum  of  partial  products  (which  begins  with  the  value  0  and  is 


gradually  built  up  to  the  complete  product)  through   dl  III   .  The 
two  inputs  t,  t'  receive  the  timing  stimuli  required  by  the  discriminator 
(they  correspond  to  t,  is  in  Figure  3.). 

7.8    The  analyri,  3f  7.7  avoi  :  I  the  f  iio\  ing  essential  fea- 
tures of  the  multiplier:   [a)  The  t  j y '  n  •  networt  which  controls- the  in- 
puts t ,  t ' ,  and  stir.ulat    ti     I  '        pr   ;r  ,"  iment  .  .  Li     vi.ll  clrar1; 
have  to  contain  j  dl   f--like   :•   r.tc  '  :f.  ' .  (0) 

-  1  dl  III 


The  k  (delay  1  >ngi  hs  )   *  ' 


a  -  1 


These  toi  be 


certain'  functions  of  synchr  nizatj  1:   F.ach  time  when  the  adder  functions 
(that  is  in  each  interval  it  -  ft   rh     tiplicand  and  the  partial  pro- 


duct sum  (that  is  the  out 


f 


md  of   di  III 


)  must  be 


brought  together  in  such  a  manner,  that    ■  former  is  advanced  by  t 
(moved  by  one   position  to  the  right)  relatively  to  the  latter,  in  com- 
parison with  their  preceding  encounter. 

Also,  if  th(  two  factors  have  30  digits  each,  the  product 
has  60  iigits.  Hence 


11  III   should  have  about  twice  the  k  of 


dl  I 


and 


dl  II 


and  a  c 


Ln  the  former  must  correspond  to  about  two 


cycles  in  the  latter.   (Th«  timing  stimuli  on  tt  will  be  best  regulated 
in  phase  with   dl  III   .)  On  the  ether  hand,  it  is  advisable  to  make 


provisions  for  rounding  I 
and  thereby  keep  the 


pre  net  off  to  the  standard  number  Iigits, 


dl  III   near  30.   (c)  The  networks  required 


dl  II 


to  get  the  multiplicand  and  the  multiplier  into   dl  I  |  and 
(from  other  parts  of  the  device),  and  to  get  the  product  out  of   dl  III 
(d)  The  networKs  required 'to  handle  the  signs  and  the  binary  point 
positions  of  the  factors.   They  are  obviously  dependent  upon  the  way  in 
which  these  attributes  are  to  be  dealt  with  arithmetically  (cf.  the  end 

-33- 


t 


: 


4 


of  7.3  and         ). 

Ail  these  points  will  be  dealt  with  subsequently.  The 
questions  connected  with  (a) — arithmetical  treatment  of  sign  and  binary 
point — must  be  taken  up  first,  since  the  former  is  needed  for  subtraction, 

I  hence  for  division  too,  and  the  latter  is  important  for  both  m»>Ttt- 
plication  and  divisl 

8.0  Circuits  for  t  h  ;_a  ••r\s  ^Jcal  operations  -.  - 

'3.1      til  now    lurr    x  ,    a  sequence  of  (about  30)  binary 
digits,,  with  no  definition  :'  s:  ;i  or  binary  point.   We  must  now  stipulate 
conventions  for  the  treatment  cf  these  concepts. 

The  extreme  Lef*  !'  1:  will  be  reserved  for  the  sign, 
so  that  its  values  0,1  exprest  the  signs  +,  -,  respectively.   If  the 
binary  point  is  between  the  digital  r  sit     1  ind  i-t-1  (from  the  left), 
then  the  positional  valre  of  the  sign  digit  is  21  .   Hence  without  the  ' 
sign  convention  the  number  x  would  lie  in  the  interval  0  ~   *  <  2- '    } 
and  with  the  sign  convention  the  subinterval  0   ~  x   <  Z  "'        is  unaffected 
and  corresponds  to  non  negative  numbers,  while  the  interv  d  ?.  ~,   X    <  Z 

corresponds  to  negative  numbers,.  We  let  the  latter  x  rej  r   nt  a  negative 
x*,  so  that  the  remaining  digits  of  x  are  essentially  the  complements  to 
the  digits  of  -x'  .   V-i-  precisely:  2.*  '/  —   (-  x    )  ■=  ~1       ,  ih.t  is 
/'*  X-  2.  *   .  To0-  t*~' %*'  <    O   .  ■ 

In  other  words:  The  digital  sequences  which  we  use 


A- 


represent,  without  the  sign  convention,  the  interval  O  =  X  <■   Z 

and  with  the  sign  convention  the  interval  -  £      =.    *         2.     .  The 

second  interval  is  correlated  to  the  first  one  by  subtracting  21  if 

-3U- 


( 


( 


necessary — that  is  their  correspondence  is  module  21. 

Since  addition  and  subtraction  leave  relations  module  *:x 
unaffected,  we  can  ignore  these  arrangements  in  ;arrying   ,1  additions 
:    ibtractions.  The  same  is  true  fcr  the    :ition  of  '.  he  binary  r-'nt:  . 
If  this  is  moved  from  i  tc  i',  then  each  :..  .  ••     '  i ..  "  1+iplie   by    "  • 

iddition  and  subtraction  leave  this  rcia'  ...  '.    -ian1  ti 
these  things  ire,  of  course,  the  analogy  c:'  '    conventional  _'  '.    pro- 
cedures. ) 

Thus  we  n  ■      not  adi  any  Lng  tc  the  additj  i       lure 
of   7»3j  and  it  will  be  ccr:  ;t  to  set  up  i   .btraction  pr  cedur<  Li1  :.he 
.  ■_  .e  way.   The  mult,.  Lies  ion  procedure    /.,',  i  a   r,  .vill  have  to  be 

nsidered,  and  the  same  caution  app'.i'.   t;  the  iivision  procedure  to 
be  set  up. 

8.2    'We  now  set  up  a  suHra^  _•  n  twork.   We  can  use  the  adder 
(cf.  7.3)  for  this  pur;cse,  if  cne  a   n  -  -  ay  the  first  one — is  fed  in 
the  negative.  According  to  the  aboy.  this  rieans  that  this  addend  x  is 
.-  ;  Laci  d  by  2X   -  x.  That  is,  each  ,i  ;.'  t   .  x  is  replaced  by  its  comple- 
ment, and  a  unit  of  th<  extr*    ri.;hc  digital  position  'a    then  add  I  to 
this  addend — or  just  as  «el]  as  ±n   extra  addend. 

This  last  operation  can  be  carried  out  by  v.xil    i 

the  extra  input  c  of  the  adder  (cf.  Figure  3.)  at  that  I  Lme.       t  ikes 
automatically  care  of  ail  carries  which  may  be  caused  by  thi:  'Xtra 
addition. 

The  complementation  of  each  iigi,t  car.  be  done  by  a 
valve  which  does  the  apposite  of  that  of  Figure  7:  When  stimulated  at 
s,  it  passes  the  complement  of  the  main  stimulus  from  is  to  os:   Figure  11. 


-35- 


i.  Bureau  of  Stand 
Division  12 
Data  Processing  Systems 


Now  the  subtracter  network  is 


P'Cr  L(7?Z       /O 


i  5 


Ol   3 


V. 


/'  /  7  :,  7?  IT      /  / 


'. 


I 


I  wn  on  Figure  11.  The  sub- 
trahend and  the  minuend  come 
in  n  '  l  e  input  lines  s,  m,  and 
the  lifference  is  emitted  with  a 
delay  3t  against  the  inputs  on 
the  output  line  d.   The  two 

inputs  t1,  t"  receive  the  necessary  timing    .  .1:  t1  thrci 
period  of  subtraction,  t"  at  its  first  t  (corresponding  to  th 
right  iigital  position,  cf.  above). 

3.3    Next   '  form  a  divider  >   >twork,  in  the  sam 
sense  as  the  multiplier  network  of  7.7. 

Binary  i vision  :  i  .< 
position  in  th<  }uo1  L  mt  (going  ;  ■ 

■. tract ed  from  the  partial  remain-!- • 
but  which  his  been  shifted  left  by  one  position,  preceding  this  subtrac- 
tion.  If  the  resulting  difference  i      r  gative  (that  is,  if  its 
extreme  left  digit  xc  7)  then  t  n      q   '.lent  digit  is  1,  and  the  next 
partial  remainder  (the  oni  to  t   ised  for  the  following  quotient  iigit, 
before  the  shift  left  referred  to  above)  if  the  differs  ce  5;  relative 
(that  is,  if  its  extreme  left  digit  is  1)  then  the  next  |u  -  i  '::i  Iigit 
is  0,  and  the  next  partial  remainder  (in  th<   ;ara   -  nse  as      I  is 
the  preceding  partial  remainder,  but  in  its  shifted  positi 

The  alternative  in  division  is  therefor*  c  nr.j  ir  ble  to 
that  one  in  multiplication  (cf.  7.7),  with  this  notable  difference:   In 
multiplication  it  was  a  matter  of  passing  or  net  passing  an  adder:,: 


if  this:   For  each  digital 
:    right) ,  the  div Lsor  is 
!   dividend)  already  formed, 


-36- 


t 


the  multiplicand,  in  division  the  question  is  which  of  two  minuends  to 
pass:   the  Shifted)  preceding  partial  remainder,  or  this  quantity  minus 
the  divisor.  Hence  we  now  need  two  valves  where  we  needed  one  in  multi- 
plication.  Also,  we  need  a  discriminator  which  is  somewhat  more  elabor- 
ate than  that  one  of  Figure  8.:   It  must  not  only  pass  a  sequence  of 
stimuli  from  is  to  os  if  there  was  a  stimulus  at  3  at  the  moment  defined 
by  the  stimulation  of  t,  but  it  must  alternatively  pass  that  sequence 
from  is  to  another  output  os'  if  there  was  no  stimulus  at  s  at  the  mo- 
ment in  question.   Comparison 


of  Figure  8.  with  Figure  12. 
shows,  that  the  latter,  posses- 
ses the  desired  properties. 
The  delay  between  is  and  os 
or  os '  is  now  3t. 


PI  $U7?£ 

'/z 

_. 

OS 

as 

/ 1 

' 

i 

r      1 

s 

■ 

s  — >— 

oC  j 

C^^—y 

—     y— — 

y 

t     t' 

- 

■^ 

I 

'-.;-.  ,-   ■  ..Now  the  divider  network  can  be  put  together:   Figure  13. 
The  divisor. cir- 


culates; through  . 


dl  I  i*  while 


the  dividend  is 
originally  in.-. 
,  but 


F  I  (,  W%£      13 


[H    our:;Yj^ 


</■- 


dl  III 


is  replaced,  as 
the  division  pro- 
gresses, by  the 
successive  partial 


A- 


rf 


oO>         JJ     I— 


*J>         &     |- 


M  *£     isr\  r 

L__. ...  _....-    ' 


M" 


i     - 


">  { 


is  \ 


:i 


i  < 


-)-j  <. 


j-  ^ 


-<—<t— <- 


A   /,  L-. 


remainders.      The   valve  f    v_^    I  routes   the   divisor  neg.itiv.ply    into  the 


-37- 


( 


f 


t 


i 


v--.adder.  The  two  valves  J  V]_  I  immediately  under  it.  select  the  partial  re- 
mainder (cf.  below)  and  send  it  from  their  common  output  line  on  one  hand 


unchanged  into 


dl  II 


and  on  the  other  hand  into  the  adder,  from  where 

thu  timing  :  :' 


dl  III 


the  sum  (actually  the  difference)  goes  into 

be  such  as  to  produce  the  required  one  position  shift  left.  Thus 

and 


dl  III 


V 


contain  the  two  numbers  from  among  which  the.  next  partial 

remainder  is  to  be  selected.  This  selection  is  done  by  the  discriminator 

* 

d,   ] Which  governs  the  two  valves  controlling  the  (second  addend)  input 
of  the  adder  (cf.  above).  The  sign  digit  of  the  resulting  sum  controls 
the  discriminator,  the  timing  stimulus  at  t  must  coincide  with  its 
appearance  (extreme  left  digit  of  the  sum),  t*  must  be  stimulated' during 
the  period  in  which  the  two  addends  (actually  minuend  and  subtrahend) 
are  to  enter  the  adder  (advanced  by  3t).   t"  must  receive  the  extra  stira- 
V   ilus  required  in  subtraction  (t"  in  Figure  11.)  coinciding  with  the  ex- 
treme right  digit  of  the  difference.  The  quotient  is  assembled  in 

,  for  each  one  of  its  digits  the  necessary  stimulus  is  avail- 


dl  IV 


able  at  the  second  output  of  the  discriminator  (os1  in  Figure  10.)  it 
is  passed  into 


dl  IV 


rl 


,  timed  by  a 


through  the  lowest  valve 
stimulus  at  t'". 

6.4    The  analysis  of  8.3  avoided  the  same  essential  features 
of  the  divider,  which  7.7  omitted  for  the  multiplier,  and  which  were 
enumerated  in  7.8: 


(a)  The  timing  network  which  controls  the  inputs  t,  tr 


t",  t'". 


(b)  The  k  (delay  lengths)  of  the)  dl  I   -  |  dl  IV 


The  details  differ  from  those  in  7.8,  (b),  but  the  problem  is  closely 
parallel. 

-38- 


( 


( 


f 


r 


-|  (c)  The  networks  required  to  get  the  divide] 


divisor  into  !  dl  III  i  and  '■    Ji  I '   .     I      ti   quo 

1 i       : I  ' 

(d)  The  networks  required  ,\    handle      '  ;ns       ary 

p  Lnt  positions. 

As  in  the  case  ci   multiplication  ali  ti  Lll 

be  dealt  >vith  subsequently. 

9.0  Th-:  binary  poi-.t 

9.1  As  pointed  out  at  the  end  of  3.1,  the  sign  convention 
. ;'  3.1  as  we'll  as  the  binary  point  convention,  which  has  not  yet  been 
determined,  have  no  influence  on  addition  and  subtraction,  but  their 
relationship  to  multiplication  and  division  is  essential  and  requires 
consideration. 

It  is  clear  from  the  definitions  of  multiplication  and 
of  division,  as  given  at  the  beginning  of  7.7  and  of  3.3  respectively, 
that  they  apply  only  when  all  numbers  involved  art  non-negative.  That 
is,  when  the  extreme  left  digit  (of  mu_tiplicand  and  multiplier,  or 
dividend  and  divisor)  is  0.  Let  us  th  -ref  re  assure  this  for  the  pr   i.1 
(this  subject  will  be  taken  up  again  in        ),  md  coarid  r  the  role 
of  the  binary  point  in  multiplication  and  division. 

9.2  As  .jointed  out  in  7.3,  (b),  the  product  of  the  30  digit 
numbers  has  60  digits,  and  since  the  product  should  be  a  iupb<  r  '.vith  the 
same  standard  number  of  significant  digits  as  its  factors,  this  necessi- 
tates omitting  30  dibits  from  the  product. 

If  the  bin-xry  point  is  betveen  the  digital  positions 

-39- 


t 


r 


i  and  i-t-  1  -     *  he- -left)  in  on-=  factor,  and  between  j  and  j-fl  in  the 
other,  then  these -nunbers  lie  between  0- and  2x~t  and  between"0  and  2J 
(the  extreme  left  digit*  isv  0,  cf.  .9-1).  iience  the  product  lies,  between  - 
0  and'21  ^~   .  However,,  if -it  is  Known  to  .lie. 'between  0  and  2*c~-'- 
(  /  =:  •■''  i  /  +  /-  /  ) >   then  its  binary  point  lies  between  k  and  k-t-1. 
Then  of  its  60  dibits  the  -first  i  +  j-l-k  (from  the  left)  are  0  and  are   ■ 
omitted,,  and  so.  it  is  only  necessary  to  omit  the  29-i- j  -fk  last  digits  • 
(to  the  right)  by  some  rounding-off  process.        '       ' 

This  shows',  that  the  essential  effect  of  the  positioning 
of  the  binary  point  is,  that  it  determines  which  digits  among  the  super- 
,  nnmerary  cnes  m  a  product  are  to  be  omitted.  > 

li   /s'.Tj-l,  then  specijj.'  precautions  must  be  taken 

k-1 

.  so  that  no  two  number?  are  ever  multiclied  for  which  the  product  is  >2 

(it' is  only  limits  bv  <21+J-2).  1  his.  difficulty  is  well  known  in 

■      ■ 
planning  calculations  on  IBLi  or  other  automatic  devices'.  There  is  an 

elegant  trick  to  get  around  this  difficulty, .due  to.  G.  Stibitz,  but  since 

it  would  complicate  the  structure  of  CA  somewhat,  we  prefer  to  carry  out 

the  first  discussion  without  -using  it .  We  prefer  instead  to  suppress 

this  difficulty  at  this  point  altogether  by  an  arrangement  which  produces 

an  essentially  equivalent  one  at  another  point.  However,  this  means  only 

-..that  in  planning  calculations  the  usual  care  •must  be  exercised,  and  it 

simplifies  the  device  and  its  discussion.  This  procedure,  'tooj'  is  in 

the  spirit  of  the  principle  of  5.6. 

This  arrangement  consists  in  requiring  k  r  1  j-1,  so 

that  every  multiplication  can  always  be  carried  out.  We  also  want  a 

fixed  position  for  the  binary  point,  coinmcn  to  all  numbers:  1  ■=  j,=  k. 

Hence  i  -'  j  *  k  =  1,  that  is:  The  binary  point  is  always  between  the 

-40- 


* 


two 'first  digital  positions  (from  the  left).   In  other 'words:  The  binary 
point  follows  always  iiamediately ''after  the  sign  "digit .  .   . 

'Thus  all  non-negative "numbers  will  be  between  0  and  1, 
ana  all  numbers  (ctf  either  sign)  between  -I  and-  1.  This  makes  it  clear 

- 

once  mbrS'  that  the  multiplication  can  always  be  carried  out. 

9.3  .  The  caution  formulated  above  is ,  therefore,  that  in 
planning  any. calculation  for  the  device,  it  is  necessary  to  sea  to  it, 
that  all  numbers. which  occur  in  the  course  of  the  calculation  should 

always  be. between  -1  and  1.  This  can  be  done  by  multiplying  the  numbers-  , 

■  ' 

of  the  actual  problem  by  appropriate  (usually  negative)  powers:  of  2 

.  (actually  in  many  cases  powers,  of  10  are'  appropriate,  cf .  •  ), 

^  and  transforming  all' formulae  accordingly.  From  the  point  of  view  of 

.  planning.it  i's  no  better  and  no  worse,  than  the- familiar  difficulty,  of 

positioning  the  decimal  point  in  most  existing  automatic. 'devices.   It  is 

necessary  to  make  certain  compensatory  arrangements  in. I  and  0,  cf .  '■ 

-  Specifically  the  requirement  that  all  numbers  remain 

between  -1  and  1,  necessitates,. to  remember  these  limitations  in  planning 

calculations:  " 

; (a)  No  addition  or  subtraction  must  be'  performed  if 

its  result  is  a  number  not  between  -1  and  l.(but  of  course  between  -2  and  .2). 

*  •  (b)  Mo  division  must  be,  performed  if  the.  divisor  is 

less  (in  absolute  value)  than  ..the  dividend. 

If  these  rulps  are  violated.,  the  adder,  subtracter  and 

divider  will  still  produce  .results,  .but  .these  will,  not  be  the' sum  diff er- 

jl   enceand  quotient  respectively.  -It  is  not  difficult  to  include  checking  . 

organs  which  signalize'  ail  infractions  of  the  rules  (a) ,  (b),  (cf.        ). 

' 


/■ 


r 


t 


I 


'A 4     "   Jn  .-junectian  with  multiplication' and  division  some  r'e- 
marks  .about  rouniin^-off  are  necessary. 

It   seems  reasonable  to   -arry  both  the^e  cperation     o-e 

digit  beyond  what   is  to  be  kept — under  ths  present  assumptions  to  %ts 

■    .         ■ 
31-st   digit— and   thrift  omit  the  supernumerary  digit  .by  sane  a^iridi'hg  pro- 
cess.'' Just  plain. .ignoring  that,  digit  would,   as  is  v.:ell  kno/ru.    c.aUs.<? 
systematical  •roundin.j-off  errors  biased  in  one.  direction   (towards  0), 
The  usual  Gaussian  decimal  procedure  of  rounding  off  to  the  nearest. value' 
of  the  laso  digit,  kept y  arid  in  case'  of  a  .(supernumerary  digit)  S  to  the  : 


V 


even  .one  means  in  tn.  binary  system  this:,'  'Digit  pairs  (3;,-st  and  31-st) 
00,10' are  rounded 'to  0,1;  '01  is  rounded  to:  00:  11  in  rounded  by  adding  01. 
This  requires  addvt Vtiii ,  with  carry  digits  and  their  inconveniences.   In- 
"Stoad  one .may  folio*  tn-;  equivalent  -.of  'the  decimal  procedure  of  rounding. 
5's  to  the  nearest  o'l   iig\i  ,  as  suggested  by  J,  ?/■,  r.auchly.  In  the 
binary  system  this  means  that  digit  pairs,  (30-st. and  31-st)  00,  01,-10, 
11  are  rounded  to  0,  1,  1,  1. 

This  rounding-off  rule  can  be  stated  very  simply:  .  The 
30-st  digit  is  rounded  to  1  if  either  the  30-'st  or  the  31-st  digit  was 
1,  otherwise  it  is  rounded  to  0. 

A  rounding-off 
valve  which  does  this  is  shown 
on  Figure  14.  A  digit  (stimu- 
lus)  is  passed  from  is  .to  os 
while  s  is  stimulated,  but  when 

s'  is  also  stimulated,  the  digit  is  combined  with  its  predecessor  (that 
is  the  ona  to  its  left)  according  to  the  above,  rounding-off  rule, 


i   -42- 


10.0  Circuit  for  the  ar:.  t  r  netical  operation  V~  .   Other  operat  ions 

10. I   A  square  rooter  network  can  bo  built  so  that  it  differs 
-  little  from  the  divider.  The  description  which  follows  is  prelim- 
inary in  the  same  sense  as  those  of  the  multiplier  and  the  divider  net- 
works in  7.7  and  3.3. 

Binary  square  rooting  consists  of  this:  For  each  digital 
position  in  the  squai  e  root  (going  from  left  to  right),  the  square  root  a 
f  ■.:    .r  to  '  ''"  •  position)  is  used  to  for;  - .■.  +  ],  and  this  2a-*- i  is 
;■■  •  ■  ;  fro.'.i  tr«  i  irti  il  remainder  (of  the  v      '.  .:       I)   air         ! 
but  ..:.;       .   ..  - :.;  ''  r  ;  left  by  two  posit  _-  •.         iv  w  dj  j'i  crs 

•  ..-■     >rigi      }      ;  ".r   exhausted),  before  this  :v.   ractior.     t  . 
res j   1  -  ■.:   .      .  •  -  negative  (that  i  .  .  L?  :J  ■     :::■■■■-■.   left  digit 

:  .•'  .  ■  .  r  root  digit  is  1  ,  .       n  ■-!  ,;  irl      mainder 
s  to  1  e  ...  •  I        following  quotie    _ .  1'  ,  before  the  loubJ 
.;  '.''  .  .•  i  f erred  to  above)  is  the  different       ;  stion.   if  the 
••  Is  negative  (that  is,  if  its  extrei   .  f1  ligit  is  I)  then 
.  luare  root  digit  is  0,  and  the  :v  .:"  ;  iri  Lai  remainder  (in  the 
as  above)  is  the  preceding  partial  remaindi  r,  but  in  its  doubly 
position. 

This  proce iure  is  obviously  very  similar  to  that  one 
1  in  division  (cf.  3.3),  with  the  follow'ing  differences:'  First:  The 
i  pie  left  shifts  (of  the  partial  remainder)  .ire  replaced  by  double  cn^:" 
(with  possible  additions  of  new  digits  0).   Second:  The  quantity  which 
is  bi  ing  subtracted  is  not  one  given  at  the  start  (the  dividend),  tut 
n>     -.ii^-    i.  determined  by  the  result  obtained  so  far:   2a+ 1  if  a  is  th<- 
squaru  root  up  to  the  position  und^r  consideration. 

-U3- 


i 


( 


The  first  difference  is  a  rather  simple  matter  of  timing, 
requiring  no  essential  additional  equipment.   The  second  difference  in- 

Lves  a  change  in  the  connection,  but  also  no  equipment.   It  is.  tru<  , 
that  2a  + 1  must  be  formed  from  a,  but  this  is  a  particularly  simple  oper- 
ation in  the  binary  system:  2a  is  formed  by  a  shift  left,  and  since 
2a  +-1  is  required  for  a  subtraction,  the  final  -+1  can  be  taken  into  ac- 
count  by  omitting  the  usual  correction  of  the  extreme  right  digit  in  sub- 
traction (cf .  8.2,  it  is  the  stimulus  on  t"  in  Figure  11.  which  is  to  be 
omitted) . 

Now 


the  square  rooter 
network  can  be  put 
together:   Figure  15 
The  similarity  with 
the  divider  network 
of  Figure  13.  is 
striking.   It  will 
be  noted  that 


di"  i 


is  not  needed.     The 


f/C7LI7?l       /£- 


*6£     I 


oU    JJ 


eC£  ZZT 


oO>   HP 


J 


t" 


v; 


0T*t 


l^y 


U 


oL. 


x      <■  <  < 


t       t 


radicand  is  originally  in  |  dl  III  1,  but  is  replaced,  as  the  square 
rooting  progresses,  by  the  successive  partial  remainders.  The  valve 

routes  the  square  root  a  (as  formed  up  to  that  position)  negatively 


into  the  adder — the  timing  must  be  such  as  to  produce  a  shift  left,  there- 
by replacing  a  by  2a,  and  the  absence  of  the  extra  correcting  pulse  for 
subtraction  (t"  in  Figures  11  and  13,  cf.  the  discussion  above)  replaces 
it  by  2a +  1.  The  two  valves  [  v  ]  immediately  under  it  select  th«e  partial 


-UU- 


r 


r 


remainder  (of.  below)  and  senH  Lt  from  their  common  output  lin 


hand  .-.changed  into  I  dl  Ii  I  and  o.i  the  other  hand  into  t  e  adder,  fr  m 
where  tne  sum  (actually  the  difference)  goes  into 


dl  III 


The  timing 


must  be  such  as  to  produce  the  required  double  position  shift  left.  Thus 


11  II  jt  and   dl  III   contain  the  two  numbers  from  among  which  the 
i i i 

next  partial  remainder  is  to  be  selected.  This  selection  is  done  by  the 
discriminator 


d^       which  governs  the  two  valves  controlling  the  (second 


addend)  input  of  the  adder  (cf.  the  discussion  of  Figure  12  in  8.3).  The 
sign  digit  of  the  resulting  sum  controls  the  discriminator,  the  timing 
stimulus  at  t  must  coincide  with  its  appearance  (extreme  left  digit  of 
the  sum)  t'  must  be  stimulated  during  the  period  during  which  the  two 
addends  (actually  minuend  and  subtrahend)  are  to  enter  the  adder  (advanced 


by  3t).   The  square  root  is  assembled  in   dl  IV  I  ,  for  each  one  of  its 
digits  the  necessary  stimulus  is  available  at  the  second  output  of  the 


discriminator  (os1  in  Figure  12),  it  is  passed  into   dl  IV   through 


the  lowest  valve  I  V]_  ,  timed  by  a  stimulus  at  t'"  . 

10.2  The  concluding  remarks  of  3.4  concerning  the  divider 
apply  essentially  unchanged  to  the  square  rooter. 

The  rules  of  9.3  concerning  the  sizes  of  numbers  enter- 
ing into  various  operations  are  easily  extended  t  j  cover  square  rooting: 
The  radicand  must  be  non  negative  and  the  square  root  which  is  produced 
will  be  non  negative.   Hence  square  rooting  must  only  be  performed  if 
the  radicand  is  between  0  and  1,  and  the  square  root  will  also  lie  be- 
tween 0  and  1. 

The  other  remarks  in  9.3  and  9.4  apply  to  square  room- 
ing too. 

10.3  The  networks  which  can  add,  subtract,  multiply,  divide 

-45- 


f 


e- 


and  square  root  having  been  described,  it  is  now  possible  to  decide  : 
they  are  to  be  integrated  in  Ca,  and  which  operations  CA  should  be  able 
to  perform. 

The  first  question  is,  whether  it  is  necessary  or  worth 
•vhile  to  include  all  the  operations  enumerated  above:  +  ,  ~,X,  f,  \f. 

Little  need  be  said  about  +,-  :   These  operations  are 

so  fundamental  and  so  frequent,  and  the  networks  which  execute  them  are 

I 

so  simple  (cf.  Figures  3  and  11),  that  it  is  clear  that  they  should  be 
included. 

With  x  the  need  for  discussion  begins,  and  at  this  'stage 
a  certain  point  of  principle  may  be  brought  out.   Prima  facie  it  would 
seem  justified  to  provide  for  a  multiplier,  since  the  operation  x  is 
very  important,  and  the  multiplier  of  Figure  9 — while  not  nearly  as 
simple  as  the  adder  of  Figure  3 — is  atill  very  simple  compared  with  the 
complexity  of  the  entire  device.  Also,  it  contains  an  adder  and  there- 
fore permits  to  carry  out  +,-  on  the  same  equipment  as  x,  and  it  has 
been  made  very  simple  by  following  the  principle  formulated  in  5r3  -  5.7. 

There  are  nevertheless  possible  doubts -about  the  strin- 
gency of  these  considerations.  Indeed  multiplication  (and  similarly 
division  and  square  rooting)  can  be  reduced  to  addition  (or  subtraction 
or  halving — the  latter  being  merely  a  shift  to  the  right  in  the  binary 
system)  by  using  (preferably  base  2)  logarithm  and  antilogarithm  tables. 
Now  function  tables  will  have  to  be  incorporated  into  the  complete  device 
anyhow,  and  logarithm — antilogarithm  tables  are  among  the  most  frequently 
used  ones — why  not  use  them  then  to  eliminate  x  (and  r, v  )  as  special 
operations?  The  answer  is,  that  no  function  table  can  be  detailed  enough 
to  be  used  without  interpolation  (this  would  under  the  conditions  contemn 

-46-. 


( 


plated,  require  2^  -  10  entries!),  and  interpolation  requires  multi- 
plication! It  is  true  that  one  might  use  a  lower  precision  multiplica- 
tion in  interpolating,  and  gain  a  higher  precision'' one  by  this  procedure-.- 
and  this  could  be  elaborated  to  a  complete  system  of  multiplication  by 
successive  approximations.   Simple  estimates  show,  however,  that  such  a 
procedure  is  actually  more  laborious  than  the  ordinary  arithmetical  one 
for  multiplication.   Barring  such  procedures,  one  can  therefore  state, 
that  function  tables  can  be  used  for  simplifying  arithmetical  (or  any 
other)  operations  only  after  the  operation  x  has  been  taken  care  of,  not  ■ 
before!  This,  then,  would  seem  to  justify  the  inclusion  of  x  among  the 
operations  of  CA. 

Finally  we  come  to  4-  and  a/ .     These  could  now  certainly 

1  V'' 

be  handled  by  function  tables:   Both  -f  and  v'  with  logarithm  -  antilogarit'hu. 

ones,  -^  also  with  reciprocal  tables  (and  x).'  There  arc'  also  well  known,  . 

2 
fast  convergent  iterative  processes:  For  the  reciprocal  u—  2u  -  au  m 

;  I 

(2  -  au)  u  (two  oDerations.x  per  stage,  this  converges  to  a),  for  the 

I  ,    I 

square  root  u  -  2  u  -  2auJ5  -  (2  -  (2au)  u)  u  (three  operations  x  per 

stace,  this  converges  to     :   .  hence  it  must  be  multiplied  by  2a  at. 

V  -ya. 

the  end,  to  give  /" a). 

However,  all  these  processes  require  more  or  less  in- 
volved logical  controls  and  they  replace  -f  and  V    by  not  inconsiderable 
numbers  of  operations  x.  Now  our  discussions  of  x,  r ,  v  show,  that  each 
one  of  these  operations  lasts,  with  30  (biaary)  digit  numbers  (cf.  7.1), 
order  of  30  t,  hence  it  is  wasteful  in  time  to  replace  t ,  -/"by  even 
a  moderate  number  of  x.   Besides  the  saving  in  equipment  is  not  very 
significant:  The  divider  of  Figure  13  exceeds  the  multiplier  of  Figure  9 


-47- 


(( 


f 


f 


by  above  50%   in  equipment,  and  it  contains  it  as  a  part  so  that  duplica- 
aions  are  avoidable.   (Cf.  ).  The  snuare  rooter  is  almost  iden- 

tical with  the  divider,  as  Figure  1$  and  its  discussion  show. 

Indeed  the  justification  of  using  trick  methods  for  -f , 
\  ,  all  of  which  amount  to  replacing  them  by  several  x,  exists  only  in 
devices  where  x  has  been  considerably  abbreviated.   As  mentioned  in  5.3  - 

5.4  the  duration  of  x  and  also  of  -$■  can  be  reduced  to  a  much  smaller  num- 

.... 
oer  of  t  than  what  we  contemplate.  As  pointed  out  loc.  cit.,  this  involves 

telescoping  and  simultaneising  operations,  and  increasing  the  necessary 
'  equipment  very  considerably.   We  saw,  that  such  procedures  are  indicated 
in  devices  with  elements  which  do  not  have  the  speed  and  the  possibilities 
of  vacuum  tubes.   Jn  such  devices,  the,  further  circumstance  may  be  impor- 
tant, that  X: can  be  more  efficiently  abbreviated  than  -r  (cf.  5.4),  and 
.  it  may  therefore  be  worth,  while  to  resort  to  the  above  mentioned  procedures, 
which  replace  ~  ,./   by  several  x.   In  a  vacuum  tube  device  based  on  the 
principles  of  5.3  -  5.7,  however,  x,  4- ,  \/ar.e  all  of  the  same  order  of 
duration  and  complication  and  the  direct  arithmetical  approach  to  all  of 
them  therefore  seems  to.  bo  justified,  in  preference  to  the  trick  methods 
discussed  above. 

Thus  all  operations  +,  -,  X,  -r ,  Vwould  seem  to  deserve 
inclusion  as  such  in  CA,  more  or  less  in. the  form  of  the  networks  of  Figures 
3,  11,  9,  13,  15,  remembering  that,  all  these  networks  should  actually  be 
merged  into  one,  which  .consists  essentially  of  the  elements  of  the  divider, 
Figure  13..  The  whole  .or,  appropriate  parts  of  this  network  can  then  be  ae- 
lected  by  the  action  of  suitably  .disposed  controlling  E-elements,  which  act 
as  valves  on  the  necessary  connections,  tq  make  it  carry  out  the  particular 
one  among  the  operations.  +.,  - ,  x  ,  v,  v  which  is  desired.  (Cf. 

For  additional  rematks  on  specific  operations  and  general  logical  control,  cf- 

-UB-  ' 


t 


10.4   The  next  question  is,  what  further  operations  (besides 
+  ,  -  ,  y  ,    4 ,  y/  )  would  be  included  in  CA? 

As  pointed  out  in  the  first  part  of  10.3  once  x  is  avail- 
able, any  ether  function  can  be  obtained  from  function  tables  with  interpo- 
lation.  (For  the  details  cf.-      ).  Hence  it  would  seem  that  beyond  x 
(and  -t-  ,  -  which  came  before  it),  no  further  operations  need  be  included 
is  such  in  CA.   Actually  -J-  ,  v'""were  nevertheless  included,  and  the  direct 
arithmetical  approach  was  used  for  them — but  here  we  had  the  excuse  that 
the  arithmetical  procedures  involved  had  about  the  same-  duration  as  those 
i  x,  md  required  an  increase  of  only  about  50%   in  equipment. 

.  Further  operations,  which  one  might  consider,  will  hardly 
jet  these  specifications.'  Thus  the  cube- root  differs  in"  its  arithmetical 
treatment  essentially  from  the  square  root,  ,as  the  latter'  requires  the 
intermediate  operation  2a-)  1  (cf.  10.1}-,  which  is  very'-simple,  particularly 
^  in  the  binary  system  while  the  former  requires  at  the  same  points  'the  in-  . 
liate  speration  yi'-  -j-  3a -J-  1  -3a  *(a+l)  +  1,  which  is  much  more  com- 
plicated, since  it  involves  a  multiplication.  Other  desirable  operations — 
i   the  logarithm,  the  trigonometric  functions,  and  their  inverses — allow 
n  hardly  any  properly  arithmetical  treatment,  in   these  cases  the  direct 
roach  involves  the  use  of  their  power  series,  for  which  the  general 
logical  control  facilities  of  the  device  must  be  adequate.  On  the  other 
hand  the  use  of  function  tables  and  interpolation,  as  suggested  above  is  in 
most  cases  more  effective  than  the  direct  power  series  approach. 

These  considerations  make  the. .inclusion  of  further  algebraic- 
al jr  analytical  operations  in.CA  unnecessary.  There  are  however  some  quite 
lementiry  operations,  which  deserve,  to  be  included  for  logical  or  organiza- 
tional reasons.   In  order  to  discuss. these  it  is  necessary  to  consider  the 
^functioning  of  CA  somewhat  more  closely,  although  we  are  not  yet  ready  to  do 
full  justice  to  the  viewpoints  brought  up  in  7..  3  and  at  the  end  of  10'.3. 

-A3- 


f 


f 


11. C   Organization  of  CA.   Complete  list  of  operations 

11. 1    As  pointed  out  at  the  end  of  10.2  CA  will  be  or- 
ganized essentially  as  a  divider,  with  suitable  controls  to  modify  its 
action  for  the  requirements  of  the  other  operations.  (It  will,  of 
course,  also  contain  controls'  for  the  purposes  enumerated  in  ?8 . )   This 
imp-lies  that  it  will  in  general  deal  with  two  real  number  variables, 
Ivhich  ?o   into  the  memory  organs  fdl*  I  ^  ,   idl  IIj    of  the  divider 
network  of  Figure  13.   [These  should  coincide  with  the  \  dl  I  I  , 

■  d  1  Uj  of  the  Multiplier,  Figure  9.'  The' square  rooter,  Figure  15, 
m  'is  no  '  d_l_  I  j  ,  but  it  makes  the  same  use  ol    dl  li]     .  The 
adder  and  subtracter  were  not  con  .ectcd  on  Figures'  2,    11.  tc  such  mem- 
ory organs,  but  they  will  have  tc  be  'when  the  organization  of  CA  is 
completed.)   3o  »ve  must  tj  ink  of  CA  as  having  tub  input  organs,  i  dl  I 
and   j dl II  \     ,  .and  of  course  one  output  organ.   (The  latter  has  not 
beer,  correlated  with  the  adder  and  subtract  or,  c'f.  above.   For  the  mul- 
tiplier it  is   idl   III  |  ,  for  the  divider  arid  square  rooter  it  is 
Idl   IV  j-  .  These  things  toe  will  have  to  le  adjusted  in  the  final  or- 
ganization'ci"  CA.)   Let  us  denote  these  two  inputs  of  CA  by  Ica  and  Jca, 
and  the  output  by  0C&  (each  of  them  wi'th  its  attached  memory  organ), 
imatically  shown  on  Figure  15. 


Now  the 
following  complex  of 
problems  must  bo  con- 
sidered:  As  mentioned 

re,  particularly  in 
2o ,    an  extensive  memory 


L.1  KL  Li 


•  a 


(Tc* 


r 


A     l^± 


M  forms  an  essential  part  of  the  device.   Since-  CA  is  the  main  internal 


-30-' 


operating  unit  of  the  device  ('..:  stores,  CC  administers,  and  I,  0  main- 
tain the  connections  with  the  outside,  cf.  the  analysis  in  2),  the 
connections  for  transfers  between  M  and  CA  are  very  important.   How  arc 
these  connections  to  be  organised? 

Ic  is  clearly  necessary  to  be  able  to  transfer  from 
any  port  of  K  to  CA,  i.e.  to  Ica,  JC{, ,  and  conversely  from  CA,  i.e.  from 

0  ,  to  any  part  of  K.   Direct  connections  between  various  parts  J    M 

d:  therefore  n:.t  seem  to  be  necessary:   It  is  always  possible  tv.  transfer 
from. one  part  uf  M  to  the  other  via  CA.   (Cf.,  however,  ) 

These  considerations  give  rise  tc  two  Questions:   First:   Is  it  nec- 
essary tc  connect  each  part  cf  M  with  both  I   and ' J cn   cr  can  this  b? 
simplified?  Second:   How  are  the  transfers  f,r  m  tile  part  of  M  to  an 
other  part  of  M  to  be  handled,  where  CA  is  only  a  through  station? 

The  first  question  can  be  answered  in  the  light  of  the 
principle  of  56.-  to  place  multiple  events  in  a  temporal  succession 
rather  than  in  ( simultaneous )  spaciM  juxtaposition.   This  means  that 
t..  .  real  numbers  which  go  frwa  M  into  I   .and  Jcrx,   will  have  to  g>. 
there  in  two  successive-  steps.  This  being  so,  it  is  just  as  well  to  route 
each  real  number  first  into  1^,  nnd  to  move  it  on  (within  CA)  from 

1  ctl   to  JCfl  when  the  n<-xt  real  number  comes  (trom  M)  into  Ica.   V/e 
restate; 

Every  real  number  coming  from  M  into  CA  is  routed  into 
Lca.  At  the  same  time  the  real  number  previously  in  Icn  is  moved  on  to 
>.Tca»  and  the  real  number  previously  is  J  is  necessarily  cleared,  i.e. 
^.forgotten-.  It  3houdd  be  noted,  thf.t  I cn  and' J  can  be  assumed  to  con- 
tain memory  organs  of  the  type  discussed  in  76.   (Cf.  Pigure  6,  there, 

-51- 


c 


cf.  also  the  varioua  fdl  '  i  in  the  x,    '/,  ,  ^networks  in  Figures*  9,  13., 
15.)  in  .vhich  the  re.^l  numbers  they  hold  ire  circulating.   Consequently 
the  connections  of  I   and  Jca  in  CA  ^rt  those  indicated 'in  Figure  17. : 
Tut  lines  -  -  -  conduct  w!.en  a  real  number  Ifrcm  M)enters  CA,  the  lines 
conduct  at  all  ether  times.   The  connections  of  I„Q  and  J   with 

,    ,         OB.  Cfl 

t 
| ■ ' 

the  operating  parts 

of  CA  are  supposed 

to  branch  out  from 

the  twe  terminals 

— •  •   Tjuj  output 

0       connects   with 
ca 

the   outside   (relatively  t;    CA,    i.e.   with  M)   by  the   line   ,   which 

Conducts  when  a   result   leaves   CA    (irr  LI)  •      The   circulating  connections 
of   0_„   and  its   connections   v/ith  the  cperating  parts    of  CA  are   not   shown, 
nor   the   E-eliments   which  control   the   connections   shown    (nor,    cf   course, 
the   operating  parts    of    CA).      (For   the   complete   description  of  CA  of. 

.) 

♦  1.2      iiuh  the   help  of  Figures    16,    17   the   second  question  is 
ilso   easily  answered.      Fur    a   transfer   from   one   part    cf   M  to   another  part 
cf  M,   going  through  CA,   the   portion  of  the  route   inside   CA  i.s  cleurly 
a  transfer   from  Icn    Jr  Jc&  to  0ca.      Denoting  the   real   numbers   in  I      , 
Jca  by  x,   y,    this   amounts   to  "combining"   x,   ;■   to   either  x   or  y,    since 
tlit:    "result"   of  any   operation  performed  by   CA    (like-)-,   -,    x,  "/'•  ,   <if~) 
is    supposed   to   uppenr  at   0      .      Tr.is   operation  is   trivial   and  a   special 
case   e.g.    of   addition:      If   s    (or  y)    ij  wanted   it   suffices   to  get   zero 
in   the   place   of  y    lor  x)   -   i.„.    into  1^   (wr  j      )   _   .md  then  M>ply   the 


-52- 


# 


€ 


operation-*-*  On  the  ether  hand,  however,  it  seems  preferable  tc  intro- 
duce these  operations  as  3uch:   Flr9t:  "Getting  eerc  int&  Icg  |ur  3^a')M 
is  u  .necessarily  tim$  consuming.  Sec  nd:   The  .direct  transfer  fr.-m 
Icn  (cr  Jcaj  tc  Oea,  which  these  operations  require  is  easily  effected 
by  a  small  part  if  the  CA  network  visualised  at  the  beginning  <£■    11.1. 
Third:   We  propose  tc;  intr<  duce  both  operations  (for  Ica  as  v;ell  as  fcr 
Jcn),  because  it  v;ill  appear  that  each  con  play  ft!  separate  useful  role  in 
the  internal  administration  f  CA  tcf.  below). 

/  introduce  accordingly  two  new' opera* ions:  i  and  j , 
corresponding  tt  direct  transfers  from  Ica  or  Jca  tc  0ca. 

These  "two  eperations  have  these  further  use's:   It  will 
be  s^en  (cf.       )  that  the  ;utput  of  CA  (from  Oca)  can  be  fed  back 
directly  int^  the  input  of  C7.  [to  Ic-t  this  ru.  ves  the  c  .ntonts  cf  I 
intc  Jca  f.nd  clears  Jca,  cf  11.1.).  Now  assume  that  lctl,   Jca  contain 
the  real  numbers  x,  y,  and  that  i  or  j  is  applied,  in  ccnjuncti<n  with 
this  feedback.  Then  the  ccnttntj  of  I   Jca  are  replaced  by  x,  x  cr 
y,  k.   I.e.  from  the  point  of  view  of  .any  wther  two  variable  operations 
(«.  -i  •/•  .  i>e«  **y»  *-y,3yt   r  )  the  variables  x,  y  have  been  replaced 
by  ;:,  x  or  y,  x.  Ik w  the  litter  is  an  important  manipulation  for  the 

V 

un symmetric  operations  (x-y,  J"  ),  and  the  former  is  important  for  the 
symmetric  -perati.ns  (x+y,  xy)  since  it  lends  tc  doubling  and  squaring. 
Beth  manipulations  are  frequent  enough  in  ordinary  algebra,  to  justify 
a  direct  treatment  by  means  cf  the  operations  i,  j. 

11.3  A  further  necessary  operati  n  is ■connected  with  the  need 
to  be  able  tc  sense  the  3ign  of  a  number,  ^r  the  order  relation  between 
two  numbers,  and  to  choose  accordingly  between  twe  (suitably  given) 


-53- 


t 


alternative   courses   ef  acti.  n.      It  will   ?.ppe'.r   later,    that   the  ability 
t.    ch'.cso   the   first   „r  the   second  en.    :1    tv,      .  iv,n  number's  u,    v,    in 
depend  net    upen  sriich  an  alternative,   is   quite    adequate   te  mediate   the 
ch.ice  between  any  fcv/i    ^iven  alternative    c  ur'sea   ef  uctlciK      (Cf. 
Acccrdingly,    we   need  an   operati   n  which   can  4.    "his:      Given  f  ^ur.  nur.- 
hers  x,   y,   u,    v,    it   "forms"    »--if  x=y.      (This    .-erases   the   order   rtlati    n 
betwetn  x,   y.      If   we   put  y  -«-0,    it    senses   the    sign  cl    :■:.) 

In  this   f-.m  the    ..per' tUn  has   f>.ur   variables:    ;:,   y, 
u,    v.      (In   th'.    sign   form  it  has   three    variables:    x,    u,    v.)      New    the 
scheme    f.  r  the    CA  net.,    rk  eh    sen  at  the    beginning  of    11.1,   which  was 
essentially   that    .  ne    c-f   the  divider,  Jiad  rcora  f .  r   twi    variables   Only, 
and  this   is  equal  Xjjf  true   f  r  the    dieeussicn   .  f  the    inputs   cf  CA   in  11*1. 
Hence    four    (er   three)    variables   '.re    t   ■    many.      Consequently   it   is    neces- 
sary t..  break  ^ur  c perati   n  up  int      tw.    variable   .  per'ti   ns  -   and  then 
we   taight   as  well   d(      his   ..  ith  the  more   general    (four  rather  than  three 
variables)    fern. 

It    is  plausible   to  be,' in  with  &■  (partial!    operation 
which  merely   dteid.'s   whether   x  '-    y  or  x<y   and   remembers    this,    but  with- 
out  taking  any  action  yet.     This    is   best    lone   by  forming  x  -  y,   and 
then  remembering  its    sif-n  di^it    only,,  i.e.    its   first  digit    (from  the 
left).      {Cf.   ai       This   digit   is   0  for  x  -  y  *   o,-i.e.   x^y,   and  it    is 
1   for  x  -  y<o,    i.e.    x'y.j     Thus   this    (partial)    operation  is  essentially 
in  the   nature   of  a   subtraction,    and   it   can  therefore   present  no  addi- 
tional difliculties   in  a  CA  which  can   subtract.      Now   it   seems  best   to 
arrange'  things   so,    that   once   this   operation  has  b/-.n  performed,   CA  will 

simply  wait   until   two  new  numbers   u,    v   have   been  moved   into  I_„,    J 

ca   ca 

(thus  clearing  t,  y  out  -  if  u,  v  are  to' occupy  Icn,,  Jca,  respectively, 

-54- 


c 


then  v  must  be  feci   in  first   and  u  second),    an     •      n   transfer    (without 
any   further   instructions,)    u  or   v   into  Cc„    (i.e.    p  norm  i   or  j)    accord- 
ing to  whethr  the   sign  digit   referred    to   above    «■  s    0   or    1. 

We   introduce  accordingly  such  nn  operation:    3.      It   is 
most   convenient    tc    arrange ~t rings  so,    that   after  x,   y  have   occup' 
I-„.    J„,- ,    a  subtraction  is   ordered  and  provisions  nu~do   that  the   result 
x  -  y   should  remain  in  0cr .     Then  x,   y  must  be   displaced  frttn  Ica>    JC!i 
by  u,    v  and  s   ordered.      3  will  sense  whether  the   number  in  Cca  is  ».o 
or  <o   (i.e.   x^   y  ,.  r  x<y)»    clear  it.  from  0ca,    n.nd  "fcrm"  accordingly 
u  or  v   in  0C%.     The   oueitition  pr . ceding  s  need,   by  the   way,   not   b<     sub- 
tr-.cti.n:      It   might   be   addition-  6r   i   c  r  j.      Accordingly   the   numb-- r   in 
0c,lf   which  provides   the   criterium  for •  5; yi  11   n.,t  be   x  -  y,   but    *i*y' 
.r  x  or  y.      I.e.    s  will   for  u  i'r"v   according,  t     whether   the  multipli- 
city r>n  or  the  division,   tnd  the  -former  might    indeed  be   sometimes  useful. 
F  r   details    .  f  those   :.peratilns   cf. 

11.-*       Combining   the e<  nclusicns   cf   10.2,    10.4,    11.2,    11.3  a 
list  of   eight   ■.  per-.t  i<  ns'  cf  CA  obtai-ns.s 

+    ,     -,     X,' ■:/.     ,  |r-,    i  ,     j,     9. 

I      t)  '-■■'     more    ..ill   have  to  be   addod,    because    .f    tho   necessity     f  ccn- 

v.rting  number.;  between  tho  binary  and  the  decimal   systems,    as   indi- 
cated at   tho  end   .  f  5.2V    Thus  we  noed  n  decimal  -   to  -  binary  conversion 
and  a   binary  -   to  -  d<  cimal'  conversions  • 

db,    bd. 
Th-.    netw   rr.s  which  carry  Cut  these  t,i.    operations  will  be  discussed 
in  . 

This    concludes   for   the  n.  ra.ent    the   discussion    -f  CA. 
We  have   enumerated  tho   ten  operations   *hich  it  must  be   at>l<  r:\rnw 

-55- 


* 


The  questions   if  7.8,,    the  general  control  problems   d     11.1,    -ind   tjj$ 
specific   networks   for  db,    bd   still   remain  t,   be   disposed    of.      But   it 
is  bettor   t.    return  tc    these   after  V5.ri.u3   other   characteristics   of 
the  device   have  been  d.  cid<  d  upon.      .Ve   postpone   therefore   their  discus- 
sion and   turn  now   tc    ^ther   parts    ,f   the   d-.vic^. 

12.0  Capacity   cf  the  mem,ry  M»        General   principles 

12.1        We   consider   next   the    third  specific   part:    the   men^ry  M. 

Memory  devices  were   discussed  in  7.5    ,    7.6    ,    since   they  are 
needed  as   ports    of    the  x,     •/.    ,      networks    (of.    7.4    ,    7.7      for   x,    83.    fur    /•  , 
10.2   fori')   and  hence    ^f   CA  itself    (df.    the    be f  inning  of   11.1    ).      In  all 
these    casts   the   devices   considered  hid  ,a   sc-qut  ntial   or  delay   chrracter, 
which  was    in  nioat   cases  made   cyclical  by   suitable   terminal   i  rgans. 

'Ire    precisely:  

The   blocks  li. 


a  nd 


dl    (k)  in  7b    ,    76     are 


essentially  delays , which   h   Id  a   stimulus   that   enters   there    input   for  a 

tine   kt ,   and  then  emit   it.      Consequently  they  can  be   converted   into 

cyclical  memories,    -Jhich  hold  a  stimulus   indefinitely,   and  make   it 

available  at   the  output  at   all  times  which  differ  from  each  other  by  h 

multiples   of  kt.      It   suffices   for   this   purpose   to-   feed   the   output   back 

into   the    input:     •/    ;_■  |l_  *\  or       >'     <  I    (  k,,      i  ~       .      Since   the   period 

kt   contains   k   fundamental  periods   t,    the   capacity   of   such  a  m'-mcry  device 

is   k   3tinuli«      The   above   schemes   l&Ck   the   proper   input,    cle'  ring  and 

output   facilities,   but   those  are    shown  on  Figure   6.      It    should  be   nrtod 

that    In  Figure  6.    tho    cycle   around        Ik  goes    thr  ugh  enc  more   E-ele- 

ment,    and   therefore   the  period   of   this  device   is   actually    lk  +  1)    t,nnd 

its   capacity  cnrrespi.nding.ly  k  rl   stimuli.      (The  !IK         -f  Figure  &• 

may,    of  course,    be   replaced  by   a  ••  ,    -41    (k)[       ,    cf.    76.) 

-56- 


t 


Now  it  is  by  no  means  necessary,  that  neu-  ry  be  f  this 
cyclical  (or  delay)  type  •   './e  must  therefore  before  making  a  decision 
concerning  M,  discuss  other  possible  types  and  the  advantages  and  dis- 
advantages of  the  cyclical  type  in  comparison  with  them. 

12.2   Preceding  this  discussion,  however,  we  must  consider 
the  capacity  which  we  desire  in  M.   »Ve  did  already  r.iontion  r.bove  this 
concept  of  capacity  for  II  or  a  part  of  U.:      It  ia  the  number  of  stimuli 
which  this  organ  can  r<.uember,  or  itfore  precise lyj  the  number  of  occasions 
for  which  it  can  remember  whether  or  not  a  stimulus  was  present.   The 
presence  or  absence  jf  a  stimulus  lat  a  ^iven  occasion-,  i.e.  on  a  given 
line  in  a  given  moment)  can  be  used  tl  express  the  value  1  or  0  fir  a 
binary  digit  (in  a  given  position)-.-  Hence  the  capacity  of  a  memory  is 

i 

the  number  uf  Mnary  digits    (the   values   ...  f)    .,/hich  it  can  retain.     In 
>. ther  words; 

The    '(capacity),  .unit   of  memory     is    (the   ability  to 
retain  the   value  of  one    binary  digit. 

vie   can  new  express  the  "cost"  of  various  types  of  in- 
fematien  in  these  nem  ry  unit^. 

Let  us   consider  first   the  memory   capacity  required  to/ 
store  a    standard-  ( real)    number.     As;. indicated  in  71-»  we   shall   fix  the 
size   of   such  a  number  at  30  binary  digits    (at   least   for  m.,st  uses,   cf. 
)     This  k-eps  the  relative  r<  unding-ef f  errors   below  2        , 
which  c   rresponds  tu^lO-9,    i.e.    t<-   carrying  y  significant  decimal  digits. 

Thus   a   standard  number   corresponds   to  20  memory,  units.      To   this  must  be 

\      ■'  !  .         . 

added     ne   unit  for  its 'sign   ( cf.   the    end _rf  92.)    and  it    is   advisable 

t     add  a  further  unit   in  lieu  of    a  symbol  which  characterises   it   as  a 


-57- 


number  ( K  distinguish  it  fr  o  an  '.rder,  cf.  ).   In  this 

*r.y  we  arrive  to  32  -  25  units  per  number. 

?he  fact  th'it  a  number  requires  32  Memory  units,  makes 
it  advis-ible  to  subdivide  the  entire  nem-ry  in  thi3  way;   First,  b- 
vieusly,  intt  units,  second  int,  fr  ups  i.f  32  units,  to  be  called 
minor  cycles.   (F.r.  the  naj  r  cycles  cf.  ) 

Each  standard  Ireal)  number  accordingly  •  .couples  precisely  ine  miner 
cycle.  It  simplifies  the  i.rgahizatUn  .  f  the  entire  memory,  and  vari- 
us  synchr- ni?nticn  pr i.tjlens  if  the  device  rilong  with  it,  if  all  other 
ponsfcanta  of  the  meau.ry  are  also  msde  tc  fit  ints.  this  subdivision  into 
mi  a  r  cycles. 

Recalling  the  classification  (a)  -  (h)  of  24'.  for  the 
presumptive  contents  cf  the  memory  M,  we  note:  (a),  iccording  to  our 
present  ideas  belongs  to  CA  and  not  to  M  (it  i9  handled  by  Jdl  I 


to   fdl  l"vj   ,  cf.  the  beginning  of  11.1)  (t )  -  (g)  ,  and  probably  (h) 
also,  consist  of  standard  numbers.   (},)  on  the  other  hand  consists  of 
the  operation  instructions  which  govern  the  functioning  of  the  device, 
to  be  .called  standard  orders.   It  will  therefore  be  necessary  to  for- 

,  mulate  the  standard  orders  in  such  a  manner,  that  each  one  should  also 

a  ■   t 

occupy  precisely  one  minor  cycle,  .'.e.  32  units.  Thi9  will  be  done  in 

<       • 
12 « 3   ,'/e  are  now  in  a  position  to  estimate  %hi    Capacity  re- 
quirements of  each  memory  type  (a)  -  (h)  of  2.4. 

Au  (a):   Need  not  be  discussed  since  it  is  taken  care 
•of, in  CA-  (df.  above).  Actually,  since  it  requires  \  cl     Ij  to  i  dl__  IV]   , 
efich  of  which  must  hold  essentially  a  standard  number,  i.e.  30  units 


-58- 


( 


f 


i 


(with  9r.i5.11  deviations,   cf.  ),    this  corresponds  to  - 

120  units.     Si^se  this   is  not  in  M,    th<.    organization  into  minor 
cycles  does  not  .apply  here,   but  we  note   that  -  120  units   correspond  - 
to  -  4  r.;  nor  cycles.      Of  course   some  other  parts  of  CA  r-re  memory 
organs   too,    usually  with  capacities  of  one   or  a  few  units:    E.g.   the 
discriminators   oi    Figures   8.   and   12.      The   complete   CA  actually 
contains  /  mere  j  ^     I  organs,    corresponding  to  -  /  units,   i.e.   -   o 
minor  cycles    (  cl'.  )  • 

Ad   (b):     The   capacity  required  for  this  purpose  can 
only  be    estim&ted    after   the   form  oi    all   standard  orders  has. been  de* 
cidtd  upon,   and  several   typical  prblems  have  ■'  been  formulated  -  '.'set 
up"  -  in  that  terminology.     This  ./ill  be  done  in  . 

It   ..ill   then  appear,    that   the    capacity   r'c  quiT-  nant's   of    (L)    are    small 
compared  t^   those   of  acne  of   (c)   -    (h),   particularly  tc    those   of.  (c). 

Ad   (c):    As  indicated   loc,    cit.,   we   c^unt   on  function 
tables   of   100-2'  £    entries.     A  function  table  is  primarily  a  switch* 

ing  pr  blen,   end  the   natural  numbers  of  alternatives  "for  a   switching 

7 
system  are   the  powers   of  2,      lCf»  ,)      Honce   126    -  2'    is  a 

suitible  number  of  entries.     Thus'  the   relative  precision  <  bt'-.ined 

—7  — "fi 

directly  for  the  variable  19  2.  ,   Since  n  relative  precisi on  of  2  vU 

is  desired  for  the  result,  and  (2-'~7)4>  2*20,  (2  7)5  «  2^°,    the-  in- 
terpolate n. error  must  be  fifth  order,  i.e.  the  interpolation  bi- 
quadratic.  (One  might  go  tc  even  higher  Order  interpolation,  and 
hence  fewer  entries  in  the  function  table.   However,  it  ..ill  appear 
that  the  capacity  requirements  of  tc)  are  even  for  128  •' ntri'es  small 
conp&red  e.  g.  tc  the  3-  of  (c)«)  With  biquadratic  interpolation  five 


-59- 


t 


( 


0 


f 


tnble    values   are   needed   for  each   interpolation:      Two  above   rmd   two 
belc*   %be    rounded  off  vnrir.ble.      Hence    128  entries  allow   actually  the 
use   of    124   only,    and  the 3c- correspond   U    122   intervals,    i.e.    n   rela- 
tive precision   122  •1   fcr  the   variable.      However  even  122-     <*'-    2-^° 

1 
(by  a  factor  -25), 

Thu3  r.   function  table   consists   cf    128  numbers,    i.e. 
it  requires  a  capacity  cf  128  minor   cycles.     The  familiar  .-mtheraati- 
cal  prtblens  hardly  ever  require  more   than  five   function  tables    (very  , 
rarely  that  much),    i.e.   a  capacity  cf  640  minor  cycles   seen  to  be  a    . 
sr»fe  ■.  verestimate    ^f  the   capacity  required  f  ^  r    (c). 

Ad   (d):      The3e  capacities  are  clearly  less  than- or 
at  most   comparable    to  those  required  by    (e).      Indeed  the    initial 
values  are   the    same   thing  as  the  intermediate   values   of    If),   except 
that   they  "belong   to  the   first   value   of  t.      And   in   a  partial,  di-ffer- 

ential  equation  with  n  +  1  variables,    say  xi   , ,   xn  and  t,-  the 

intermediate   values   of  a  given  t   -   to  be   discussed   under    (e)   -   as 
.well-  as  the   initial  values   or  the   totality  of    all   boundary  values 
for  all   t  correspond  all  three  to  n-dimensional  manifolds    (in  the  n  + 

1  -   dimensional      space)    of  x\    ,   ,   *n  and  t;    hence   they   are  likely 

to   involve   all  about   the   same  number  of  data. 

Another   important  point    is,    that   the    initial   values 
and   the  boundary  values  are  usually  given  -  partly  or  wholly  -  by  a 
formula  -   or  by  a  moderate  number  of  formulae.      I.e.,    unlike  the 
intermediate   values  of    (e),   they  need  not  be   remembc>red  as   individual.. 
numbers. 

Ad  •(e):      twr  a  partial   differential    equation  with 
two  variables,   say  x  and   t,    the   number  of  intermediate   v.lues   for  a 

-60- 


given  t   is   d-.termined  by  the  number  of   x  -   lattice  points  used   in  the 
calculation.     This    is  hardly   ever  more   thin  150,    and   it    is   unlikely 
that  more   than  5  numerical  quantities    should   be   associated   with   each 
point. 

Ixi  typical   hydrodynaraical   problems,    ./here   x   iB  «■ 
Lagrange ian  label   -   coordinate,    50-100  points   are   usually   a   light 
estimate.,    and  2  numbers  are   required  at  each  point;    A  fosition  -  co- 
ordinate  and  a  velocity.     Returning  to  the  higher  est imr.te   of   150  points 
and  5  numbers  at  each   point  gives   750  numbers,    i.e.    it   requires  a 
capacity  of   750  minor   cycles.      Therefore    1,000. minor  cycles   seen  to 
be  a   safe  overestimate   of  the    capacity  required  for   (e)    in  two  vari- , 
able   9x  and  t)   prcbler.s. 

For  a  partial   differential   equation  with   three   vari- 
ables,   say  x,   y   and  t,    the  estimate    is   harder  to  make.      In  hydro- 
dynamical  problems,   at   least,   important  progress  could   be  uo.de  with 
20  x  20  or  40  x'2C   or   similar  numbers  of  x.,  y  -   lattice  points  -   say 
1,0^0  points.      Interpreting  x,   y  again  in  Lo<.  range  i  an  labels   3hows, 
that   at   least   4  numbers   are   neeaed  at   each  point:      Two  position  co- 
ordinates and   two  velocity  components,      we   take  6  numbers  per  point 
to  allow   for   possible   other  m.n  hydrodynaraical  quantities.     This   gives 
6.CCC  numbers,    i.e.    it  requires  a  c  u-city  of   6,000  minor  cycles   for 
(e)    in  hydro  dynamical   three    variable    (x,   y  and  t)    problems. 

It   will   be   seen   (cf.  ),   that  A   memory 

capacity    .f   6,010  min^ r  cycles  -   i.e.    of   -   200, 0CC   units  -   is   still 
conveniently  fe-siblt  but   that   essentially  higher  capacities  would 
be   increa3infly   difficult   to   control      Even  200,000  units   produce   some- 


-61- 


( 


.vh".t  „f  *.n  unbalance  -  i..e.  they  n-.ke  M  bigger  than  the   thejr  parts 
i.  f  the  device  put  t^£-ether.   I-  seems  r.herefcre  un-vise  tt  gt  further, 
-;nd  tc  try  t~  tret  for  variable  i;:,  y,  z  and  t)  pr.bler.is. 

It  should  be  noted  that  two  variable  (x  and  t)  prob- 
leras  include  all  linear  or  circular  symmetric  plane  or  spherical  sym- 
metric snaciul  transient  problems,  also  certain  general  plane  or 
cylinder  symmetric  SDacial  stationary  problems  (they  must  be  hyper- 
bolic, e.g.  supersonic,  t  is  replaced  by  y).  Three  variable  problems 
{:•:,  y  and  t)  include  all  spacial  transient  problems.  Comparing  this 
enumeration  with  the  well  known  situation  of  fluid  dynamics,  elasticity, 
etc.,  shows  how  imoortant  each  on<?  of  these  successive  stages  is: 
Complete  fret  dom  with  two  variable  problems^  extension  to  four  variable 
Drobiems.   «.s  we  indicated,  the  possibilities  of  the  practical  size 
for  'iA   draw  the  natural  limit  for  the  device  contemplated  at  present' 
between  the  second  and  the  third  alternatives.  It  will  be  seen  that 
considerations  of  duration  place  the  li:nit  in  the  same  place  (cf.       ) 

nd   (fji   The-memory  capacities  required  by  a  total 
differential  equation  with  two  variables  -  i.e.  to  the  lower  estimate 
of  (e). 

Ad  (g):  As  pointed  out  in  (g)  in  2k.,    these  problems 
ire  very  similar  to  tr.ose  of  (e),  except  that  the  variable  t  now  dis- 
arm. Henc<  '.he  lower  estimate  of  (e)  (1,000  minor  cycles)  applies 
when  a  system  of  (at  nost  5)  one-varii.ble  functions  (of  x)  is  being 
p^ht  by  successive  •iDnroxii:'.---tion  or  relazation  methods,  while  the 
u.er  estimate  of  (e)  (6,000  minor  cycles)  applies  when  a  system  of 
•-■  st  6)  two-variable  functions  (of  x,  y)  is  bein£  sought.  Lany 


-62- 


c 


( 


problems  of  this -type,  however,  deal  with  one.  function  only  -  this 
cuts  the  above  estimates  considerably  (to  2>-l0  or  1,000  minor  cycles). 
Problems  in  which  only  a  system  of  individual  constants  is  being  sought 
by  successive-  aoproxinat ions,  require  clearly  smaller  capacities:  '  They 
compare  to  the  preceding  Droblems  like  (f)  to  (e). 

ad  (h);   These  problems  are  so  manifold,  that  it  is 
difficult -to  plan  for  them  systematically  at  this  stage. 

In  sorting  problems,  any  device  not  based  freely 
oermutable  record  elements  (like  puncheards)  has  certain  handicaps 
(cf.  ),  besides  this  subject  can  only  be  adequately  treated 

z't  r  an  analyst  of  the  relation  of  l.l   and  of  R  has  been  made  (cf.  29 
and       ).  It  should  be  noted,,  however,  that  the  standard  punchcard 
has  place  for  80  decimal  digits.,,  i.e.,  -  1     9-cigit  decimal  numbers, 
th ;t  is  -  9  numbers  in  our  present  S' nse,  i.e.  -  9  m  nor  cycles.  Hence 
'.he  6,000  minor  cycles  considered  in  (e)  correspond,  to  a  sorting  capa- 
city  of  -  700  fully  used  cards.   In  the  most  sorting  problems  the  80 
columns  of  the  cards  a  far  from  fully  used  -  this  may  increase  the 
equivalent  sorting  capacity  of  pur  device  proportionately  above  700. 
This  means-,  that  the  device  has  a  non  negligible,  but  certainly  not 

: ssi-ve  sorting  capacity.  It  is  probably  only  worth  using  on  sorting 
problems  of  more  than  usual  mathematical  complexity. 

In  statistical  experiments  the  memory  requirements  are 

usually  small:   Each  individual  Droblem  is  usually  of  moderate  com- 

i 

plexity,    each' individual  nroblera  is  independent   (or  only  dependent  by 
a  few  data)  from  its  •predecessors;   and  all  that  need  be  remembered 
through  the  entire^  sequence  of  individual  orobi'-ms   are  the  numbers  of 


.--63- 


c 


1 


( 


( 


how  many  problems   successively  solved  had  their  results  in  each  one  of 
a  moderate  number  of  p.iven  distinct  classes. 

12. A     The  estimates  of  12.3  can  be   summarized  as  follows: 
The  needs  of  (d)  -   (h)   are  alternative,    i.e.  they  cannot  occur  in  the 
sane  problem.     The  highest   estimate  reached  hero  was  one  of  6,000  rain or 
cycles,   but  already  1,000  nunor  cycles  would  p'ermit  to  treat  many  im- 
portant probl-ras.      (a)  need  not  be  considered  in  A.      (b)   and  (c)  a 
cumulative,   i.e.    they  may  add  to  (d)  -   (h)   in  the   same  problem.      1,000 
minor  cycles  for  each,    i.e.   2,000  together,    seem  to  be  a  safe  over- 
estimate.    If  the  higher  value   6,000  is  used   in  (d)  -   (h),   these  2,000 
may  be  added  for  (b)  -  (c).     If  the  lower  valuo  1,000  is  used  in  (d)  - 
(h),    it   seems,  reasonable  to  cut   the   (b)  -   (c)   capacity  to  1,000  to. 
(This   amounts  to  ass^imii\g  fewer  .function  tables  and  somewhat  less 
•complicated   "set  ups" .     Actually  even  those  estimates  are  generous, 
cf.  )     Thus  tot cil  capacities    jf  8,000  or  2,000  minor  cycles 

obtain. 

It  will  be  seen  that  it  is  desirable  to  have  a  capa- 
city of  minor  cycles  which   is. a  power  of  two  (cf.  ).     This 
makes  the  choices  of  8,900   jr  2,X0  monor  cycles  of  a  convenient  approxi- 
mate size:      They  lie  very  near  to  powers  of, two.     We  consider  accord- 
ingly" those  t..o  t  <tal  memory,  capacities:    8,196  ~  2^  or  2,0LB  *  2" 
:..inor   cycles,    i.e.   262,272 ■=  218  or  65,336  =  216  units.     For  the 
purposes  of  the  discussions  which  follow  we   will  use  the  first  higher 
■-  a  1 1  mat  e . 

This  result  deserves  to  be  noted.      It  shows  in  a  most   strik- 
ing  way  whore   t,  he,.,  real  difficulty,   the  main  bottleneck  of  an  automatic 

-6v 


< 


i 


very  high  st  cd  computing  device  lies!  At  the  memory.   Compared  t< 
the  relative  simplicity  of  CA  (cf.  the  beginning  of  11.1  and        ), 
and  to  the  simplicity  of  CC  and  of  its  "-code"  (cf.        iftd       ), 
!,i  is  somewhat  impressive:   The  rcnuirerients  formulated  in  12.2,  which 

were  considerable  but  by  no  means  ohantastic,  necessitate  a  memory  M 

- 
with  a  capacity  of   about  a  quarter  -dllion  units',   Clearly  the 

practicality  of  u  device  as  is  contemplated  here  depends  most  critically 

on  the  possibility 'of  building  such  an  ii,  and  on  the  question  of  how 

simple  8uch  an  ivi  can  be  made  to  be. 

12 .5   How  can  an  M  of  a  capacity  of  -  218 250*000  units 

be  built? 

The  necessity  of  introducing  del-y  elements  of  very 

great  efficiency,  as  indicated  in  75'.,    76.  \    and  12.1,  becomes  now 

obvious:'  One  iv-element,  as  shown  oh  Figure  /„'. ,  has  &   unit  memory 

.•■.'''•.a'         ■  » 

capacity,  hence  any  direct  solution  of  the  problem  of  construction  'd 

with  the  help  of  it-elements  would  require  as  many  E-eloments  -as  the 
desired  capacity  of  1-i  -  indeed,  because  of  the  necessity  of  switch- 
ing and  gating  about  four  times  morej  (cf.        )'.   This  is  m?ni- 
festly  impractical  for  the  desired  capacity  of  -  25u',000  -  or,  for 
that  natter,  for  the  lower  alternative  in  12.5',  of  -  65, OCX). 

We  therefore  return  to  the  discussion  of  the  cyclical 
or  delay  memory,  which  was  touched  upon  in  12.1.  (An  other  type  will 
be  considered  in  12.6) 

Del      fdl  (k)|    can.  be  built  with  great  capacities 
k,  without  using  any  L-e  laments  at  all.  This  was  mentioned  in  76, 
together  with  the  fact  that  even  linear  electric  circuits  of  this  type 


-65- 


. 


exist'.   Indeed,  the  contemplated  t-  of  about  one  microsecond  requires 
a  circuit  passband  of  3  -  5  megacycles  (remember  Figure  1.  '.  )  and 
then  the  equipment  required  for  delays  of  1  -  3  microseconds  -  i.e. 
k  =  1,  2,  3  -  is  simple  and  cheap,  and  that  for  delays  up  to  30  -35 
microseconds  -  i.e.  k  =30, ,  35  -  is  a-vailable  and  not  unduly  ex- 
pensive or  complicated.   Beyond  this  order  of  k,  however }   the  linear 
electric  circuit  approach  becomes  impractical. 

This  means  that  the  delays  — >*-   >^ — .  ???—   which 
occur  in  all  ^-networks  of  Figures  3.  -15.  can  be  easily  made  with 
linear  circuits',  also,  that  the  various  Jdl  |   of  CA  (cf.  Figures  9, 
13,  15,  and  the  beginning  of  11. 1( ,  which  should  have  k  values  -  30, 
and  of  which  onlv  a  moderate  number  will  be  ne-'-ded  (of.  (a)  in  12.3), 
can  be  reasonably  made  with  linear  circuits.  For  k   itself,  however, 
the  situation  is  different. 


ii  must  be  made  un  of  j  dl    organs,  of  a  total 
capacity  -  250, OuO.'  If  these  were  linear  circuits,  of  maximum  capa- 
city -  3U  (cf.  abovo),  then  -  3,000  such  organs  would  be  required, 
which  is  clearly  impractical.  This  is  also  true  for  the  lower 

it  .  rna'ive  of  12  .'5 ,  capacity  -  65,000,  since  even  then  -  2,000  such 
organs  would  be  n   cessary. 

i'OW  it  is  possible  to  build   dl   organs  which  have 
an  electrical  innut  :ind  output,  but  not  a  linear  electrical  circuit  in 
between,  idth  k  values  up  to  several  thousand.  Their  tv  tar-.'  is  such, 
that  a  -  U   stage  amplification  is  needed  at  the  output,  which,  apart 
from  its  anpiifying  character,  also  serves  to  reshape  and  resynchronize 
the  output  pulse."  I.e.*  the  last  stage  gates  the  clock  pulse  (cf.  63.) 

-66- 


( 


( 


—  using  a  non  linear  pert  of  a  vacuum  tube  characteristic  which  goes 
across  the  cutoff;  while  all  other  stages  effect  ordinary  amplification, 
using  linear  parts  of  vacuum  tube  characteristics.  Thus  each  one  of 
these  J  dl  I  requires  -  U   v  cuum  tubes  at  its  output,  it  also  requires 
-  4  E-elements  for  switching  and  gating  (cf.        ).  This  gives 
probably  10  or  fewer  v. cuum  tubes  per  i  dl  i  organ.  The  nature  of 
these  dl   organs  is  such,  that  s    few  hundred  of  them  can  be  built 
and  incorporated  into  one  device  without  undue  difficulties  -  although 
they  will  then  certainly  constitute  the  greater  oart  of  the  device 
(cf.        ). 

Uow  the  a   cap-  city  of  25U,O00  can  be  achieved  with  such 
dl   devices,  each  one  having  a  capacity  1,000  -  2,000,  by  using 
250  -  125  of  them.   Such  numbers  are  still  manageable  (cf..  above),  and 
they  require  about  8  times  more,  i.e.  2,500  -  1,250  vacuum  tubes. 
This  is  r.i  considerable  but  perfectly  practical  number  of  tubes  -  in- 
deed probably  considerably  lower  than  the  upper  limit  of  practicality. 
Th'  fact  that  they  occur  in  identical  groups  of  10  is. also  very  ad- 
vantageous.  (For  details  cf.         )   It  will  be  seen  that  the 

j 
other  parts  of  the  device-  of   which  CA  aid  CC  aire  electrically  the 

most  complicated,  renuir;  together  <^1,000  vacuum  tubes.  (CL.  ) 

Thus  the  vacuum  tube  requirements  of  the  device  are  controlled  essen- 
tially b"  *i,  and  'hey  ire  of  the  order  of  2,000  -  3/000.   (Cf.  loc. 
cit.  -'.'rove.)   This  ccyifirms  the  conclusion  of  12. U,,    that  the  decisive 
cart  of  t'r  .  device,,  determining  more  than  -iny   other  part  its  feasi- 
bility, dimensions  and  cost,  is  the  memory. 


-67- 


■. 


i 


I 


We  must  ncW  decide  more  accurately  what  the  caoocity 
of  each  ,  dl    organ  should  be  -  within  the  linits  which  were  found  to 
bo  pr  c-ic.l.  h   combination  a  few  very  simple  viewpoints  loads  to 
such  .  decision. 

12.6   v/c  saw  above  that  each   I  dl  j   organ  requires  about 
10  .ssoci;  ted  vacuum  tubes,  essentially  ind<  -x-ndently  of  its  length, 
(A  very  long  i  dl  ]   night  require  one  more  stage  of  amplification,  i.e. 
il  vacuum  tubes.)  Thus  the. number  of  i  dl  i   organs,  and  not  the 
total  capacity  determines  the  number  of  vacuum  tubes  in  M.  This  would 
justify  using  as  few   dT]   organs  as  possible,  i.e.  of  as  high 
indivicu-:  1  cap:>city  as  nossible,  'Now  it  would  probably  be  feasible  to 


devi  I'd  '(  dl  i  's  of  the  type  considered  with  capacities  considerably 
higher  th'n  the  few  thousand  mentioned  above.  There  are,  however, 
r  consid  orations  which  set  a  limit  to  increases  of  j  dl 

In  tlit  first  place,  the  consid'  rations  at  the  end  of 


63.  "row,  that  the  definition  of  j  dl   's  deiav  tjjae  must  be  a  frac- 
tion t"  '.  :  t  (  .bout 5  ~T"),  so  that  -.a ..eh  stimulus  emerging  from  \  dl  ) 
may  gate  the  correct  clock  oulse  for  the  output.   For  :.  capacity  !:, 
i.e.  a  delsy  kt,  this  is  relative  precision  5k  -2k,  which  is  perfectly 
feasible  for  the  device  in  question  when  k  -  1,000,  but  becomes  in- 
creasingly uncertain  when  k  increases  beyond  10,000.  However,  this 
argument  is  limited  by  the  consideration  that  as  the  individual  J dl  | 
capacity  increases,  corrusoondinply  fewer  such  organs  are  needed,  -aad 
therefore  each  one  can  be  Made  <«ith  corr<  spondngly  more  attention  <*nd 
precision. 


-63- 


£ 


( 


K. 


Z 


i 


Next,  there  is  another  more  sharply  limiting  consider- 
ation.  If  each  i   dl~~  ;  has  the  capacity  k,  then  ^0,000  of  them  will 
be  needed,  an  ^0|QP°  amplifying  switching  and  gating  vacuum  tube  ag- 
gregates  are  necessary.  Without  going  yet  into  the  details  of  these 
circuits,  the  individual  |  dl  )  and  its  associated  circuits  can  be  shown 


f / q  ure       / i 


"L 


cdt 


^y 


schematically  in  Figure  18. 

Mote,  that  Figure  6.  showed 

the  block  SG  in  detail  but 

the  block  A  not  at  all. 

The  actual  arrangement  will 

differ  from  Figure  6.  in 

some  details,  even  regarding  SG,  cf.  .  Since  \    dl  \  is  to  be 

used  as  a  memory  its  output  must  be  fed  back— directly  or  indirectly— 

into  its  input.  In  an  aggregate  of  many  |  dl  |  organs— which  M  is  going 

to  be— we  have  a  choice  to  feed  each  j  dl  \  back  into  itself,  or  to  have 

longer  cycles  of  i  dl   j's:  Figure  19.  (a)  and  (b),  respectively. 


F  I  q   (/  r  e. 


*J 


-na 


>-[ 


eU 


vt   | L2£j— * 


^O 


J — Q— illh 


It  should  be  noted,  that  (b)  shows  a  cycle  which  has  a  capacity  that 
is  a  multiple  of  the  individual  j  dl   |' s  capacity— i.e.  this  is  a  way 


to  produce  a  cycle  which  is  free  of  the  individual |  dl j's  capacity 

limitations.  This  isr  of  course,  dua  to  the  reforming  of  the  stimuli 


-69- 


c 


f 


v  traversing  this  aggregate  at  each  station  A.  The  information  contained 
in  the  aggregate  can  be  observed  from  the  outside  at  every  station  SG, 
and  it  is  also  here  that  it  can  be  intercepted,  cleared,  and  replaced   | 
by  other  information  from  the  outside.   (For  details  cf .  ) 

Both  statements  apply  equally  to  both  schemes  (a)  and  (b)  of  Figure.  19. 
Thus  the  entire  aggregates  has  its  inputs,  outputs,  as  well  as  its 
switching  and  gating  controls  at  the  stations  SG — it  is  here  that  all 
outside  connections  for  all  these  purposes  must  be  made. 

^)  To  omit  an  SG  on  the  scheme  (a)  would  be  unreasonable: 

It  would  make  the  corresponding   dl   complete  inaccessible  and  use- 
less. In  the  scheme  (b),  on  the  other  hand,  all  SG.  but  one  could  be 
omitted  (provided  that  all  A  are  left  in  place):  The  aggregate  would 
still  have  at  least  one  input  and  output  that  can  be  switched  and  gated 
and  it  would  therefore  remain  organically  connected  with  the  other  parts 
of  the  device — the  outside  in  the  sense  used'  above. 

We  saw  in  the  last  part  of  12.5,  that  each  A  and  each 
SG  required  about  the  sane  number  of  vacuum  tubes  (4),  hence  the  omission 
of  an  SG  represents  a  $0%   saving  on  tho  associated  equipment  at  that 
junction. 

Now  the  number  of  SG  stations  required  can  be  estimated. 
(It  is  better  to  think  in  terms  of  scheme  (b)  of  Figure  19  in  general,  and 
to  turn  to  (a)  only  if  all  SG  are  known  to  be  present,  Cf.  above.)  Indei  d: 
Let  each   dl   have  a  capacity  k,  and  let  there  be  an  SG  after  every  1 
of  them.  Then  the  aggregate  between  any.  two  SG  has  the  capacity  k'  =  kl. 
(One  can  also  use  scheme  (b)  with  aggregates  of  1   dl   's  each  and  one 


v 


3G  each.)  Hence  2,5,° i.QPQ  SG's  are  needed  altogether,  and  the  switching 

-k' 

-70- 


<c 


problem  of  M  is  a   A  » way  one.   On  the  other  hand  every  individual 

K 

emory  unit  passes  a  position  SG  only  at  the  end  of  each  k't  period. 
I.e.  it  becomes  accessible  to  the  other  parts  of  the  device  only  then. 
Hence  if  the  information  contained  in  it  is  required  in  any  other  part 
of  the  device,  it  becomes  necessary  to  wait  for  it — this  waiting  time 
being  at  most  k't,  and  averaging  \   k't. 

This  means  that  obtaining  an  item  of  information  from 
M  consumes  an  average  time  \   k't.  This  is,  of  course,  not  a  time  re- 
quirement per  memory  unit:  Once  the  first  unit  has  been  obtained  in 

^'  this  way  all  those  which  follow  after  it  (say  one  or  more  minor  cycles) 
consume  only  their  natural  duration,  t.  On  the  other  hand  this  variable 
.-.  .  Lting  time  (maximum  k't,  average  5  k't),  must  be  replaced  in  most 
cases  by  a  fixed  waiting  time  k't,  since  it  is  usually  necessary  to 

^_  return  to  the  point  in  the  process  at  which  the  information  was  desired, 
after  having  obtained  that  information — and  this  amounts  altogether  to 
a  precise  period  k't.   (For  details  cf.  ).   Finally, 

this  wait  k't  is  absent,  if  the  part  cf  M  in  which  the  desired  information 

'3?    is  contained  follows  immediately  upon  the  point  at  which  that  information 
is  wanted  and  the  process  continues  from  there.  We  can  therefore  say: 
The  average  time  of  transfer  from  a  .general  position  in  M  is  k't. 

Hence  the  value  of  k'  must  be  obtained  from  the  general 
principles  of  balancing  the  time  requirements  of  the  various  operations  of 
the  device.  The  considerations  which  govern  this  particular  case  are  simplt 

In  the  process  of  performing  the  calculations  of  mathe- 
matical problem  a  number  in  U   will  be  required  in  the  other  parts  of  the 
device  in  order  to  use  it  in  some  arithmetical  operations.   It  is  excep- 
"/  lional  if  all  these  operations  are  linear,  .i.e.  +  ,  -  ,  normally''-,  and 

-71- 


C'/ 


\c 


( 


possibly  "f ,  v  ,  will  also  occur.   It  should  be  noted  that  substituting 
^  a  number  u  into  a  function  f  given  by  a  function  table,  so  as  to  form 
f(u),  usually  involves  interpolation — i.e.  one  x  if  the  interpolation 
is  linear,  which  is  usually  not  sufficient,  and  two  to  four  x's  if  it 
is  quadratic  to  biquadratic,  which  is  normal.   (Cf.  e.g.  (c)  in  12.3.) 
A  survey  of  several  problems,  which  are  typical  for  various  branches  of 
computing  mathematics,  shows  that  an  average  of  two  x  (including  -f  ,  v  ) 
per  number  obtained  from  M   is  certainly  not  too  high.   (For  examples 
cf . .        )  Hence  every  number  obtained  from  M  is  used  for  two 
multiplication  times  or  longer,  therefore  the  waiting  time  required  for 
obtaining  it  is  not  harmful  as  long  as  it  is  a  fraction  of  two  multipli- 
cation times. 


^ 


9 

A  multiplication  time  is  of  the  order  of  30  times  t 
(cf.  5.3,  7.1  and. 12.2,  for  -f,  v~cf.  5-5)  say  1,000  t.  Hence  our 
condition  is  that  k't  must  be  a  fraction  of  2,000  t.  Thus  k'— 1,000 

with  k — 1,000  is  perfectly  feasible 


seems  reasonable.  Now  a 


dl 


(cf.  the  second  part  of  12.5),  hence  k  «  k' — 1,000,  1  -   1  is  a  logical 


choice.  In  other  words:  Each   dl  has  a  capacity  k — 1,000  and  has 
an  SG  associated  with  it,  as  shown  on  Figures  18.,  19. 

This  choice  implies  that  the  number  of   dl   j's  re- 
quired. isr~  3/V iff 0°  —  250  and  the  number  of  vacuum  -tubes  "in  their  asso- 
ciated  circuits  i's  about  10  times  more  (cf.  the  end  of  12.5.),  i.e. — 
2,500. 

12.7   The  factorization  of  the  capacity— 250,000  into 


250 


dl  J  organs  of  a  capacity — 1,000  each  can  also  be  interpreted  in 


this  manner:   The  memory  capacity  250,000  presents  prima  facie  a  250,000 — 


-72- 


C' 


(' 


or 


«<£ 


way  switching  problem,  in  order  to  make  all  parts  of  this  memory  immed- 
'    lately  accessible  to  the  other  organs  of  the  device.   In  this  form  the 
task  is  unmanageable  for  E-elements  (e.g.  vacuum  tubes,  cf.  however  12.8) 
The  above  factorization  replaces  this  by  a  250 -way  switching  problem, 
and  replaces  for  the  remaining  factor  1,000  the  (immediate,  i.e.  syn- 
chronous) switching,  by  a  temporal  succession — i.e.  by  a  wait  of  1000  t.  . 

This  is  an  important  general  principle:  A  c  =  hk  - 
way  switching  problem  can  be  replaced  by  a  k  -  way  switching  problem 

and  an  h-step  temporal  succession  -  i.e.  a  wait  of  ht.  We  had  c  - 

'n  ■■ 
y     250,000  and  chose  k  -  1,000,  h  -  250.  The  size  of  k  was  determined  by 

K. 
the  desire  to  keep  h  down  without  letting  the  waiting  time  kt  grow 

beyond  one  multiplication  time.  This  gave  k  -  1,000,  and  proved  to  be 


^ 


dl 


of  capapity  k. 


compatible  with  the  physical  possibilities  of  a 

It  will  be  seen,  that  it  is  convenient  to  have  k,  h, 
and  hence  also  c,  powers  of  two.  The  above  values  for  these  quantities 
are  near  such  powers,  ,and  accordingly  we  choose: 


Total  capacity  of  M: 

c  ::  262,144  =  2XH. 

Capacity  of  a. i  dl  J' organ: 

k  =   1,024  =  210. 

Number  of   dl  J  organs  in  M: 

ha     256  =  28. 

The  two  first  capacities- are  stated'in  memory  units. 
In  terms  of  minor  cycles  of  32  -  2  memory  units  each:"" 

Total  capacity,  of  1!  in  minor  cycles: c/32  -  8,192  =  2  -*. 

Capacity  of  a '._dl  ; organ  in  minor  cycles:  k/32  a    32  =  2  . 

12.8   The  discussions  up  to  this  point  were  based  entirely  on 
the  assumption  of  a  delay  memory.   It  is  therefore  important  to  note  that 
this  need  not  be  the  only  practicable  solution  for  tne" memory  problem  - 
indeed,  that  there  exists  an  entirely  different  approach  which  may  even 

-73-' 


(>r 


" 


«; 


appear  prima  facie  to  be  more  natural. 
\%  The  solution  to  which  we  allude  must  be  sought-  along 

the  lines  of  the  iconoscope.  This  device  in  its  developed  form  remembers 
the  state  of  -  400  x  500  =.  200,000  separate  points,  indeed  it  remembers 
for  each  point  more  than  one  alternative.  As  it  is  well  known,  it  re- 
members whether  each  point  has  been  illuminated  or  not,  but  it  can  dis- 
tinguish more  than  two  states:   Besides  light  and  no  light  it  can  also 
recognize— at  each  point — several  intermediate  degrees  of  illumination. 
These  memories  are  placed  on  it  by  a  light  beam,  and  subsequently  sensed 

by  an  electron  beam,  but  it  is  easy  to  see  that  small  changes  would  make 

**'  ...'■'■••■■■ 

it  possible  to  do  the  placing  of  the  memories  by  an  electron  beam  also. 

Thus  a  single  iconoscope  has  a  memory  capacity  of  the 

same  order  as  our  desideratum  for  the  entire  M  (-250,000),  and  all 

memory  units  are  simultaneously  accessible  for  input  and  output.  The 

situation  is  very  much  like  the  one  described  at  the  beginning  of  12.5., 

and  there  characterized  as  impracticable  with  vacuum  tube-like  E-elements, 

The-  iconoscope  comes  nevertheless  close  to  achieving  this:   It  stores 

200,000  mem6ry  units  by  means  of  one  dielectric  plate:  The  plate  acts 

in  this  case  like  -200, 000  independent  memory  units — indeed  a  condenser 

is  a  perfectly  adequate  memory  unit,  since  it  can  hold  a  charge  if  it 

is  properly  switched  and  gated   (and  it  is  at  this  point  that  vacuum  tubes 

are  usually  required).     The  250,000-way  switching  and  gating  is  done   (not 

by  about  twice  250,000' vacuum  tubes,  which  would  be  the  obvious  solution, 

but)  by  a  single  electron  beam — the  switching  action  proper  being  the 

steering  (deflecting)  this  beam  so  as  to  hit  the  desired  point  on  the 

plat  e  i  ' 


e 


( 


r 


e* 


% 


Nevertheless,  the  iconoscope  in  its  present  form  is  not 
immediately  usable  as  a  memory  in  our  sense.  The  remarks  which  follow 
bring  out  some  of  the  main  viewpoints  which  will  govern  the  use  of 
equipment  of  this  type  for  our  purposes. 

(a)  The  charge  deposited  at. a  "point"  of  the  icono- 
scope plate,  or  rather  in  one  of  the  elementary  areas,  influences  the 
neighboring  areas  and  their  charger.  Hence  the  definition  of  an  elemen- 
tary area  is  actually  not  quite  sharp.  This  is  within  certain  limits 
tolerable  in  the  present  use  of  the  iconoscope,  which  is  the  production 
of  the  visual  impression  of  a  certain  image.  It  would,  however,  be 
sntirely  unacceptable  in  connection  with  a  use  as  a  memory,  as  we  are 
contemplating  it,  since  this  requires  perfectly  distinct  and  independent 
registration  and  storage  of  digital  or  logical  symbols.   It  will  prob- 
ably prove  possible  to  overcome  this  difficulty  after  an  adequate  devel- 
opment— but  this  development  may  be  not  inconsiderable  and  it  may  neces- 
sitate reducing  the  number  of  elementary  areas  (i.e.  the  memory  capacity ) 
considerably  below  250,000.  If  this  happens,  a  correspondingly  greater 
number  of  modified  iconoscope  will  be  required  in  U. 

(b)  If  the  iconoscope  were  to  be  used  with  400  x   500  = 
200,000  elementary  areas  (cf.  above),  then  the  necessary  switching,  that 
is  the  steering  of  the  electron  bean  would  have  to  be  done  with  very  . 
considerable  precision:   Since  500  elementary  intervals  must  be  distin- 
guished in  both  directions  of  linear  deflection,  a  minimum  relative 
precision  of  — -  x  t-t-t  =  ,1%   will  be  necessary  in  each  linear  direction. 
Titis  is  a  considerable  precision,  which  is  rarely  and  only  with  great 
difficulties  achieved  in  "electrical  analogy"  devices,  and  henco  a  nost 

-75- 


€ 


(  i 


1 


fe 


;-m    inopportune  requirement  for  cur  digital  device.  A  more  reasonable,  but 
still  far  from  trivial,  linear  precision  of,  say,  ,5£  would  cut  the  '  ' 
memory  capacity  to  10,000  (since  100  x  100  ■=  10,000,  -i-  x  -i-  ='  .5%)'. 

There  are  ways  to  circumvent  such  difficulties,  at  least 
in  part,  but  they  cannot  be  discussed  here. 

(c)  One  main  virtue  of  the  iconoscope  memory  is  that 
it  permits  rapid  switching  to  any  desired  part  of  the  memory.   It  is 
entirely  free  of  the  octroyed  temporal  sequence  in  which  adjacent  memory 
units  emerge  from  a  delay  memory.  Now  while  this  is  an  important  advan- 

Y 

tage  in  some  respect,  the  automatic  temporal  sequence  is  actually  desirable 
in  others.  Indeed,  when  there  is  no  such  automatic  temporal  sequence  it 
is  necessary  to  state  in  the  logical  instructions  which  govern  the  prob- 
lem precisely  at  which  location  in  the  memory  any  particular  item  of 
information  that  is  wanted  is  to  be  found.  However,  it  would  be  unbear- 
ably wasteful  if  this  statement  had  to  be  made  separately  for  each  unit 
of  memory.  Thus  the  digits  of  a  number,  or  more  generally  all  units  of 
a  minor  cycle  should  follow  each  other  automatically.  Further,  it  is 
usually  convenient  that  the  minor  cycles  expressing  the  successive  steps 
in  a  sequence  of  logical  instructions  should  follow  each  other  automat- 
ically. Thus  it  is  probably  best  to  have  a  standard  sequence  of  the  con- 
stituent memory  units  as  the  basis  of  switching,  which  the  electron  beam 
follows  automatically,  unless  it  receives  a  special  instruction.  Such 
a  special  instruction  may  then  be  able  to  interrupt  this  basic  sequence, 
and  to  switch  the  electron  beam  to  a  different  desired  memory  unit  (i.e. 
point  on  the  iconoscope  plate). 

This  basic  temporal  sequence  on  the  iconoscope  plate 
corresponds^  of  course,  to  the  usual  method  of  automatic  sequential  scan- 
ning with  the  electron  beam — i.e.  to  a  familiar  part  of  the  standard 

-76- 


i 


< 


k 


i 


..-^  iconoscope  equipment.  Only  the  above  mentioned  exceptional  voluntary 
switches  to  ether  points  require  new  equipment. 

Tc  sum  .:p:  It  is  not  the  presence  of  a  basic  temporal 
sequence  of  memory  units  which  constitutes  a  weakness  of  a  delay  memory 
as  compared  to  an  iconoscope  memory,  but  rather  the  inability  of  the 
former  to  break  away  from  this  sequence  in  exceptional  cases  (without 
paying  the  price  of  a  waiting  time,  and  of  the  additional  equipment 
required  to  keep  this  waiting  time  within  acceptable  limits,  cf.  the 
last  part  of  12.6  and  the  conclusions  of  12.7)-  An  iconoscope  memory 

V 

should  therefore  conserve  the  basic  temporal  sequence  by  providing  the 
usual  equipment  for  automatic  sequential  scanning  with  the  electron 

un,  but  it  should  at  the  same  tine  be  able  of  a  rapid  switching  (de- 
flecting)  of  the  electron  beam  to  any  desired  point  under  special  in- 
^   struction. 


(i)  The  delay  organ   dl   contains  information  in  the 
form  of  transient  waves,  and  needs  a  feedback  in  order  to  become  a  (cy- 
clical) memory.  The  iconoscope  on   the  other  hand  holds  information  in 
a  static  form  (charges  on  a  dielectric  plate),  and  is  a  memory  per  se. 
Its  reliable  storing  ability  is,  however,  .not  unlimited  in  time — it  is 
a  matter  of  seconds  or  minutes,  './hat  further  measures  does  this  neces- 
sitate? 

It      I  b  noted  that  M's  main  function  is  to  store 
information  which  Ls  required  while  a  problem  is  bring  solved,  since  it 
is  then  that  there  is  a  .ue     ■  the  rapid  accessibility,  which  the 
main  advantage  of  M   over  outside  storage  (i.e.  over  h,  cf.  2.9).  Longer 
range  storage--e.g.  of  certain  function  tables  like   log,  sin,  or 


-77- 


<r 


I 


( 


r 


€ 


..-^  equations  of  state)  or  of  standard  logical  instructions  (like  interpola- 
tion rules)  b       problems,  or  of  final  results  until  they  are  printed — 
should  be  definitely  effected  outside  (i.e.  in  h,  cf.  again  2.9  and      ) 
Hence  LI  should  only  be  used  for  the  duration  of  one  problem  and  consider- 
ing the  expected  high  speed  of  the  device  this  will  in  many  cases  not  be 
long  '/riough  tc   '  .'  ct  th<  reliability  of  '..'..      In  some  problems,  however, 
it  will  be  too  long  (cf.  ),  and  then  special  measures  become 

necessary.' 

The  obvious  solution  is  this:   Let  Nt  be  a  time  of 
reliable  storage  in  the  iconoscope.   (Since  Nt  is  probably  a  second  to 

A         Q 

15  minutes,  therefore  t  -  one  microsecond  gives  H  -  10  -  10.  For  N,- 

10'  this  situation  will  hardly  ever  arise.)  Then  two  iconoscopes  should 

be  used  instead  of  one,  so  that  one  should  always  bo  empty  while  the 

other  is  in  use,  and  after  N  periods  t  the  latter  should  transfer  its 

information  to  the  former  and  then  clear,  etc.   If  M  consists  of  a 

'reater  number  of  iconoscopes,  say  k,  this  scheme  of  renewal  requires 

k  +  1,  and  not,  k  iconoscopes.   Indeed,  let  IQ,  I,, ,  I,  be  f:  ■ 

iconoscopes.   Let  -'  3  given  moment  I-    be  empty,  and  I, ,  I.  ■,,  I-s+is 

,  lu   in  use.   Aft.  r  -^-  periods  t  I..,  should  transfer  its  informa- 

k+1  lTi 

tion  to  I-  and  then  clear  (for  i  =  k  replace  i+i  by  0).   Thus  1^^.^ 

takes  over  the  role  of  I-.  Hence  if  we  begin  with  Ic,  then  this  process 

goes  through  a  complete  cycle  I-,,  I2, ,  Ik  anL!  back  to  *o>  in  ^  r  ^ 

steps  of  duration  -JL  t  each  i.e.  of  total  duration  Nt.  Thus  all  I0, 

k  '-i 

I,,  ,  Ik  are  satisfactorily  renewed.   A  more  detailed  [dan  jf  these 

arrangements  would  have  to  be  bas-  d  on  a  knowledge  of  the  precise  orders 
of  magnitude  of  N  and  k.   ,'.'  i   I  not  do  this  here.   We  only  witsh  to 


3 


-73- 


{ 


I 


6 


<- 


§ 


•f^i    emphasize  this  point:      All  these  considerations  bring  a  dynantical  and 
cyclical  element    into  the  use  of    t!       intrinsically  static  iconoscope — 
it  forces  us  to  treat  them  in  a  mani  er   soi     vj     I  rable  to  t!  nner 

)  which  a  delay    (cyclical  memory  treats   fc  le     ing]      memory  units. 

From   (a)   -    (d)    ..      '  -  iclude  this:      It   is   vary  probable 


that   in  the  end  the   iconoscope  memorj    rill  prove    superior  t 


'  <  a   -j 


memory.   However  this  may  recjuire  ?  >n  :   further  ic-v  J  .  > ...  nt  ii  se-v  ral 
respects,  and  for  various  reasons  the  actual  use  of  the  iconoscope 
^v   ory  will  not  be  as  radically  lifferent  from  that  of  t  i  Lay  ;  enory,  as 
•  ight  at  first  think.   Indeed,  (c)  ana  (d)  show  thai  the  two  havi 
i   deal  in  common.   For  these  reasons  it  seem:  reasonable  to  continue 
analysis  on  th.e  oasis  of  a  delay  memory  although  the  importance  of 
iconoscope  memory  is  fully  realize  i. 

13»0  Or;-, :iiz ". tion  ci  '1 

13»I       We  return  to  the  discussion  of  a   ielay  m  mi  vy  bas>  d  on 
3?)        the  analysis    and  the   conclusions  of   12. t      :. a  12.7.      It   is  b    ?t    I  :>     t  :.rJ 

by  considering  Figure  19  again,   and  the  alt  . r  .  3  which   it     r.hibits. 

a 
We  know  from  12.7  that  we  must  think  in  t  ,rms  cf  256    -  2"'      •■    ns  1       '1 

1  0 
of   capacity  1.024  -  2   "    each..     For   ?.  w]  il      it  will  nor.  be  neces:  arj    to 

lecide  which  of  the   two   alternatives  Figur*    19   ('.' )   and   (b)    (or  >.hich 

combination  of  both)  will  be  used.      (For  the  decision  of 

Consequently- we  can  replace  Figure  19  by  fcht  :    r  Figure  13. 

The  next    taak  is,    then,    •-■".    discuss  the        n  Lnal   org 

A  and  SG.     A  is  a  4  stagi      m;  .  Lfier,    about  which  more  whs   said  .5. 

^)       The  function  of   \  is   solely  to   restore   the   pu]  from 

-79-' 


> 


I 


( 


r 


c 


to  the  shape  and  intensity  with  which  it  originally  entered  i  dl   ]. 
Hence  it  should  really  be  considered  a  part  off  '  dl 


proper,  and  there 


is  no  occasion  to  analize  it  in  terms  of  E-elements.  SG,  on  the  other 
hand,  is  a  switching  and  gating. organ  and  we  should  build  it  up  from 
E-elements.   We  therefore  proceed  to  do  this. 

13.2   The  purpose  of  SG  is  this:  At  those  moments  (i.e. 
periods  ~  )  when  other  parts  of  the  device  (i.e..  OC,  CA  and  perhaps  I,  0) 


are  to  send  information  into  the  I  dl   to  which  this  SG  is  attached,  or 


when  they  are  to  receive  information  from  it,  SG  must  establish  the  . 
necessary  connections — at  such  moments  we  say  that  SG  is  on.  .At  those 
moments  when  neither  of  these  things  is  required,  SG  'must  route  the 


output  of  its   dl  j  back  into  the  input  of  its  (or  its  other)  1  dl 


according  to  the  approximate  alternative  of  Figure  19.  at  such  moments 
we  say  that  SG  is  off.  In  order  to  achieve  this  it 'is  clearly  necessary 


to  take  two  lines  from  C  (and  1,0)  to  this  SG:  .  One, to  carry  the  j  dl 


output  to  C,  and  one  to  bring  the  j  dl  |  input  from  C.  .  Since  at  any 
given  time  (i.e.  period  7  )  only  one  SG  will  be  called  upon  for  these 
connections  with  C,  i.o..beon; (remember  the  principle  of  5.6!)  there 
need  only  be  one  such,  pair  of  connecting  lines,  which  will  do  for  all 
256  SG's.  We  denote  these  two  lines,  by  LQ  and  Lj_,  respectively.  Now 
the  scheme  of  Figure  18  can  be  made  more  detailed,  as  shown  in  Figure  20. 

As  indicated,  « 0  is  the. line  connecting  the  outputs  of 
all  SG' s  to  C ,  and  /7.-Y  './  R  E       Z  U 

Lj_  is  the  line  con- 


necting G  to  the 
inputs  o£   all  SG's. 
When  SG  is  off,  its 


*jC 


— i  ft 


-16. 


sr; 


TI-±_ 


-30- 


( 


' 


€ 


connections  o,  i  with  L  ,  Lj_  are  interrupted,  its  output  goes  to  a, 


this  being  permanently  connected  to  the  input  c  of  the  proper  )  dl 


according  to  Figure  19.,  (a)  or  (b).  When  SG  is  on,  its  connections 
with  a  are  interrupted,  its  output  goes  through  o  to  LQ  and  so  to  C, 
while  the  pulses  coming  from  C  over  Lj  go  into  i  which  is  now  connected 
with  a,  so  that  these  stimuli  get  now  to  a  and  from  there  to  the  pro- 
dl  (input  (cf.  above).  The  line  s  carries  the  stimuli  which 


per 


put  SG  on  or  off — clearly  each  SG  must  have  its  individual  connection 
s  (while  LQ,   L^  are  common.) 

13.3   Before' we  consider  the  E-network  of  SG,  one  more  point 
must  be  discussed.  We  allowed  for  only  one  state  when  SG  is  on,  whereas 
there  are  actually  two:  First,  when  SG  forwards  information  from  M  to 
C,  second,  when  SG  forwards  information  from  C  to  ;U.      In  the  first  case 
the  output  of  SG  should  be  routed  into  L0,  and  also  into  a,  while  no  L^ 
connection  is  wanted.   In  the  second  case  L^  should  be  connected  to  a 
(and  hence  to  the  proper   dl 


input  by  the  corresponding  permanent 

connection  of  a).  This  information  takes  away  the  place  of  the  infor- 

.  .   mat ion  already  in  M,  which  would  have  normally  gone  there  (i.e.  the 

output  of  SG  which  would  have  gone  to  a  if  SG  had  remained  off),  hence 

the  output  of  SG  should  go  nowhere,  i.e.  no  L  connection  is  wanted. 

(This  is  the  process  of  clearing.  For  this  treatment  of  clearing 

cf .  )  To  sum  up:  Our  single  arrangement  for  the  on  state 

differs  from  what  is  needed  in  either  of  these  two  cases.   First  case: 

a  should  be  connected  to  the  output  of  SG,  and  not  to  L^.  Second  case: 

a  should  lead  nowhere,  not  to  L  .  • 

o 

Both  maladjustments  are  easily  corrected.   In  the  first 

~\       case  it  suffices  to  connect  L0  not  only  to  the  organ  of  C  which  is  to 

-81- 


(  ' 


£ 


t 


£ 


^  receive  its  information,  but  also  to  Lj_ — in  this  manner  the  output  of 

SG  gets  to  a  via  L0,  the  connection  of  LQ  with  L. ,  and  L^.  In  the  second 
case  it  suffices  to  connect  L0  to  nothing  (except  its  i's) — in  this  manner 
the  output  of  a  goes  into  L0,  but  then  nowhere. 

In  this  way  the  two  above  supplementary  connections 
of  L  and  L.  precise  the  originally  unique'' on  state  of -SG  to  be  the  first 
or  the  second  case  described  above.  Since  only  one  SG  is  on  at  any  one 
time  (cf .  13-2)  these  supplementary  connections  are  needed  only  cnce. 
Accordingly  we  place  them  'into  C,  more  specifically  into  CC,  where  they 
clearly  belong.   If  we  had  allowed  for  two  different  on  states  of  SG 
itself,  then  the  it  would  have  been  necessary  to  locate  the  E-network, 
which  establishes  the  two  corresponding  systems  of  connections,  into 
SG.  Since  there  are  256  SG's  and  only  one  CC,  it  is  clear  that  our 
*   present  arrangement  saves  much  equipment. 

I3./4.   'We  can  now  draw  the  E-network  of  SG,  and  also  the  E-net- 
work in  CC  which  establishes  the  supplementary  connections  of  L0  and  h±' 
discussed  in  13.3. 

Actually  SG  will  have  to  be  redrawn  later  (cf.       ), 
we  now  give  its  preliminary  form:  SG-'-  in  Figure  21..  When  s  is  not  stim- 
ulated the  two  (2)   are  impassable  to 
stimuli,  while  f"^  is,  hence  a  stim- 
ulus entering  at  b  goes  on  to  a,  while 
0  and  i  are  disconnected  from  b  and  a. 
When  s  is  stimulated  the  two  (2) 
become  passable,  while  f~^)    is  blocked, 

hence  b  is  now  connected  to  o  and  i  to  a.  Hence  SG"1"  is  on  in  the  sense 

I  .    '    . 

of  13.2  while  s  is  stimulated,  and  it  is  off  at  all  other  times.  The 

triple  delay  on  (^_J)  is  necessary  for  this  reason:  When  SG  is  on,  a 


-82- 


<" 


t 


i 


c 


fiQuR-E    IZ 


t 


f~& 


Y 


SCL 


TV 


L.         i. 


stimulus  needs  one  period  ?    to  get  from  b  to  o,  i.e.  to  L  (cf.  13.3  and 

the  end  of  this  section  13.4),  and  one  to  get  from  L^,  i.e.  from  i  (cf. 

■ 
Figure  20),  to  a — that  is,  it  takes  3  T  from  b.to  a.  It  is  desirable 

that  the  timing  should  be  the  same  when  SG-*-  is  off,  i.e,  when  the  stim- 
ulus goes  via  (_)  from  b  to  a — hence  a  triple  delay  is  needed  on  \~J . 

The  supplementary  connections  of  L0  and  L^  are  given  in  ' 
Figure  22.  When  r  is  not  stimulated  the  two'  \^_)  are  passable  to  stimuli; 
while  {2)  is  not,  hence  a  stimulus 
entering  at  L0  is  fed  back  into  L^ 
and  appears  also  at  C^,  which  is 
supposed  to-  lead  to  C .  .When  r  is 
stimulated  the  two  f)  are  blocked, 
while  (2\  becomes  passable,  hence 
a  stimulus  entering  at  C0,  which  is 
supposed  to  come  from  C,  goes  on  to  Lj_,  and  L0  is  isolated  from  all  con- 
nections. Hence  SCL  produces  the  first  state  of  13.3  when  r  is  not  stim- 
ulated, and  the  second  state  when- r  is  stimulated.  We  also  note,  that  in 
the  first  case  a  stimulus  passes  from  L0  to  Lj_  with  a  delay  £•  (cf .  the 
timing  question  of  SG^,  discussed  above.) 

13.5   We  must  next  give  our"  attention  to  the  line  s  of  Figure 
20. and  21:  As  we  saw  in  the  first  part  of  13.4,  it  is  the  stimulation 

of  s  which  turns  SG  on.  Hence,  as  was  emphasized  at  the  end  of  13.2, 

" 

each  SG  must  have  its  own  s — i.e.  there  must  be  256  such  lines  s.  Turn- 
■■  ing  a  desired  SG  on,  then,  amounts  to  stimulating  its  s.  Hence  it  is  at 
this  point  that  the  —250-way — precisely  256-way — switching  problem  com- 
mented upon  in  12.7  presents  itself. 

1 

- 

More  precisely:   It  is  to  be  expected,  that  the  order  to  turn 
on  a  certain  3G — say  'Ao,      K —  will  appear  on  two  lines  in  CC  re- 

-83- 


e-j 


< 


( 


t 


. .  I  for  this  purpose,  in  this  manner:   On-  stimulus  on  the  first' 
line  expresses  the  presence  of  the  order  as  such,  while  a  sequence  of  ■ 
stimuli  on  the  second  line  specifies  the  number  k  desired,  k  runs  over 

256  values,  it  is  best  to  choose  these  as  0,  1,  ,  255,  in  which  case 

k  is  the  general  8-digit  binary  integer.  Then  k  will  be  represented  by 
a  sequence  of  8  (possible)  stimuli  on  the  second  line,  which  express 
(by  their  presence  or  absence),  in  their  temporal  succession,  k*s  binary' 
digits  (1  or  0)  from  right  to  left.  The  stimulus  expressing  the  order 
as  such  must  appear  on  the  first  liner,  (cf.  above)  in  some  definite  time 
relation  to  these  stimuli  on  the  second  line— as  will  be  seen  in        , 
it  comes  immediately  after  the  last  digit. 

Before  going  on,  we  note  the  difference  between  these 
3  (binary)  digit  integers  k  and  the  30  (binary)  digit  real  numbers 
(lying  between  0  and  1,  or,  with  sign,'  between  -1  and  1),  the  standard 
•^   real  numbers  of  12.2.  That  we  consider  the  former  as  integers,  i,'e* 

with  the  binary  point  at  the  right  of  the  8  digits,  while  in  the  latter 
the  binary  point  is  assumed  to  be  to  the  left  of  the  30  digits,  is 
mainly  a  matter  of  interpretation,   (cf .  )  Their 

difference  in  lengths,  however,  is  material:  A  standard  real  number 
constitutes  the  entire  content  of  a  32  unit  minor  cycle,  while  an  8  digit 
k  is  only  part  of  an  order  which  makes  up  such  a  minor  cycle, 
(cf.  ) 

U.O   CC  and  M 

1A.1   Our  next  aim  is  to  go  deeper  into  the  analysis -'of 'CC. 
Such  an  analysis,  however,  is  dependent  upon  a  precise  knowledge  of  the 
system  of  orders  used  in  controlling  'the 'device,  since  the  function  of 

-84- 


I 


c 


e>j 


CC  is  to  receive  these  orders',  to  interpret  them,  and  then  either  to 
carry  them  out,  or  to  stimulate  properly  those  organs  which  will  carry 
•  them  out.  It  is  therefore  our  immediate  task  to  provide  a  list  of  the  . 
orders  which  control  the  device,  i.e.  to  describe  the  code  to  be  used 
in  the  device,  and  to  define  the  mathematical  and  logical  meaning  and 
the  operational  significance  of  its  code  v.ords. 

Before  we  can  formulate  this  code,  we  must  go  through 
some  general  considerations  concerning  the  functions  of  CC  and  its  re- 
lation to  M. 

The  orders  which  are  received  by  C  come  from  li,   i:e. 
from  the  same  place  where  the  numerical  material  is  stored,   (cf.  2.4 
and  12.3  in  particular  (b).)  The  content  of  M  consists  of  minor  cycles 
(cf.  12.2  and  12,7 )>  hence  by  the  above  each  minor  cycle  must  contain  a 
distinguishing  mark,  which  indicates  whether  its  is  a  standard  number 
or  an  order. 

The  orders  which  CC  receives  fall  naturally  into  these 
four  classes:   (a)  Orders  for  CC  to  instruct  CA  to  carry  out  one  of  its 
ten  specific  operations  (cf.  11.4).   (b)  Orders  for  CC  to  cause  the 
transfer  of  a  standard  number  from  one  place  to  another.   (c)  Orders 
for  CC  to  transfer  its  own  connection  with  U   to  a  different  point  in  M, 
with  the  purpose  of  getting  its  next  order  from  there,   (d)  Orders 
controlling  the  operation  of  the  input  and  the  output  of  the  device  (i;e; 
1  of  2.7  and  0  of  2.3) 

Let  us  now  consider  these  classes  (a)  -  (d)  separately; 
We  cannot  at  this  time  add  anything  to  the  statements  of  11.4  concerning 
(a),   (cf.  however  )  The  discussion  of  (d)  is  also  better 


'• 


-85- 


( 


c 


i 


^  delayed  (cf.  ).  We  propose,  however,  to  discuss  (b) 

and  ( c )  now . 

14.2   Ad  (b):  These  transfers  can  occur  within  L',  or  within 
CA,  or  between  M  and  CA,  The  first  kind  ;an  always  be  replaced  by  two 
operations  of  the  last  kind,  i.e.  all  transfers  within  11   can  be  routed 
through  CA.  We  propose  to  do  this,  since  this  is  in  accord  with  the 
general  principle  of  5.6..   (cf,  also  the  riiscussion  of  the  second  ques- 
tion in  11.1),  and  in  this  way  we  eliminate  all  transfers  of  the  first 
kind-  Transfers  of  the  second  kind  are  obviously  handled  by  the  oper- 
ating controls  of  CA,  Hence  those  of  the  last  kind  alone  remain.  They 
■  fall  obviously  into  two  classes.:  Transfers  from  M  to  CA  and  transfers 
from  CA  t,o  IL     V/e  may  break  up  accordingly  (b)  into  (b1)  and  (b" ), 
corresponding  to  these  two  operations, 
"  ILj. . 3   Ad  (c):  In  principle  CG  should  be  instructed  after 

e$ch  order,  where  to  find  the  next  order  that,  it  is  to  carry  out.  We 
saw,  however,  that  this  is  undesirable  per  se,  and  that  it  should  be 
reserved  for  exceptional  occasions,  while  as  a  normal  routine  CC  , should 
obey  the  orders  in  the  temporal  sequence,  in  which  they  naturally  appear 
at  the  output  of  the  DLA  organ  to  which  CC  is  connected,   (cf.  the 
corresponding  discussion  for  the  iconoscope  memory,  (c)  in  12.8)  There 
must,  however,  be  orders  available,  which  may  be  used  at  the  exceptional 
occasions  referred  to,  to  instruct  CC  to  transfer  its  connection  to  any 
other  desired  point  in  M.  This  is  primarily  a  transfer  of  this  connec- 
tion to  a  different  DLA  organ  (i.e.  a   dl  {'organ  in  the  sense  of  12.7) 
Since,  however,  the  connection  actually  wanted  must  be  with  a  definite 
minor  cycle,  the  order  in  question  must  consist  of  two  instructions: 


'• 


-86- 


c 


I 


i 


< 


First,  the  connection  of  CC  is  to  be  transferred  to  a  definite  DLA  organ. 
"^  Second,  CC  is  to  wait  there  until  a  definite   -period,  the  one  in  which 
the  desired  minor  cycle  appears  at  the  output  of  this  DLA,  and,CC  is  to 
accept  an  order1  at  this  time  only. 

Apart  from  this,. such  a  transfer  order' might  provide, 
that  after  receiving  and  carrying  out  the  order  in  the  desired  minor  cycle  ^ 
CC  should  return  its  connection  to  the  DLA  organ  which  contains  the  minor 
cycle  tfrnt  fallows  upon  the  one  containing  the  transfer  order,  wait  until 
this  minor  cycle  appears  at  the  output,  and  then  continue  to  accept  or- 
ders from  there  on  in  the  natural  temporal  sequence-  Alternatively,  after 
receiving  and  carrying  out  the  order  in  the  desired  minor  cycle,  CC  should 
-tinuo  with  that  connection,  and  accept  orders  from  there  on  in  the 
iral  temporal  sequence.   It  is  convenient  to  call  a  transfer  of  the 

~^   first  type  a  transient  one,  and  one  of  the  second  type  a  permanent  one. 

&  '.••■..  '  '■ 

It  is  clear  that  permanent  transf ers  are  frequently 

needed,  hence  the  second  type  is  certainly  necessary.  Transient  trans- 
fers are  undoubtedly  required  in  connection  with  transferring  standard 

fc>'      b" 
numbers  (orders  (c1)  and  (c"),  cf.  the  end  of  1A.2  and  in  more  detail 

in  1A..4  below).   It  seen:!  very  doubtful  whether,  they  are  over  needed  in 
true  orders,  particularly  since  such  orders  constitute  only  a  small 
part  of  the  contents  of  U   (cf.  (b)  in  12.3),  and  a  transient  transfer 
order  can  always  be  expressed  by  two  permanent  transfer  orders.  We  will 
therefore  make  all  transfers  permanent,  except  those  connected  with 
transferring  standard  numbers,  as  indicated  above. 

1U.L       Ad  (b)  again:   Such  a  transfer  between  CA  and  a  defi- 
nite minor  cycle  in  M  (in  either  direction,  corresponding  to  (b1 )  or 
J    (b"),  cf.  the  end  of  1/+.2)  is  similar  to  a  transfer  affecting  CC  in  the 

-87- 


( 


i 


C 


v 


sense  of  (o),  sifice  it  reauires  establishing  a  connection  with  the 
.desired  DLA  organ,  and  then  waiting  for  the  appearance  of  the  desired 
minor  cycLe  at  the  output.  Indeed,  since  only  one  connection  t-tw;  :, 
U  and  CO  (actually  '  or  CA,  i.e.  C)  is  possible  at  one  time,  such  a 
number  transfer  .  :  r<  3  abandoning  the  present  'connection  cf  CC  with 
M,  and  t,h  lg  a  new  connection,  exactly  as  if  a  transfer 

affecting  CC  in  the  cerise  of  (c)  were  intended,  Since,  however,  ac- 
tually  no  such  transfer  of  CC  is  desired,  the  connection  of  CC  with  its 
original  DLA  organ  :nust  be  reestablished,  after  the  number  transfer  has 

n  carri-  !  rat,  and  the  waiting  for  the  proper  minpr  cycle  (that,  one 
.  blowing  in  Lhfe  natural  temporal  sequence  upon  the  transfer  order)  ,ie 
also  necessary.   I.'  .  this  is  a  transient  transfer,  as  indicated  at  the 
.  of  IL .     . 

It  should  he  noted,  that  during  a  transient  transfer 
the  place  of. the  minor  cycle  which  contained  the  transfer  order,  must 
be  remembered,  since  CC  will  have  to  return  to  its  successor.   I.e. 
CC  must  b&   able  to  remember  the  number  of  the  DLn  organ  which  contains 
this  miner  cycle,  and  the  number  off  periods  after  which  the  minor 
eye.:-:  will  appear  at  the  output.   (cf.  for  details  .) 

la. b       Some  further  remarks: 

Fir;  i  :   Every  permanent  transfer  involves  waiting  for 
the  iesired  minor  cycle,  i.e.  in  the  average  for  half  a  transit  through 
.LA  organ,  512  periods  J    .  A  transient  transfer  involves  two  such 

,  which  add  up  exactly  to  one  transit  through  a  DLA  organ, 
1,02/*  periods  T  .  One  might  shorten  certain  transient,  transfers  by 
appropriate  timing  tricks,  but  this  seems  inadvisable,  at  least  at  this 
„  1        -f  the  discussion,  since  the  switching  operation  itself  (i.e. 

-88- 


changing  the  connection  of'  CC)  may  consume  a  nonnegligible  fraction  of 
a  minor  cycle  and  may  therefore  interfere  with  the  timing. 

Second:  It  is  sometimes  desirable  to  make  a  transfer 
from  M  to  CA,  or  conversely,  without  any  waiting  time.   In  this  case  the 
minor  cycle  in  M,  which  is  involved  in  this  transfer,  should  be  the  one 
immediately  following  (in  time  and  in  the  same  DLA  organ)  upon  the  one 
'containing  the  transfer  order.  This  obviously  calls  for  an  extra  type 
)f  immediate  transfers,  in  addition  to  the  two  types  introduced  in  14.3. 
This  type  will  be  discussed  more  fully  in 

Third:  The  256  DLA  organs  have  numbers  0,  1,  ,  255, 

i.e.  all  8-digit  binary  numbers.  -It  is  desirable  to  give  the  32  minor 
cycles  in  each  DLr.  organ  equally  fixed  numbers  0,  1,  — -,  31,  i.e.  all 
5-digit  binary  numbers.  Now  the  DLA  organs  are  definite  physical  objects, 
h^nce  their  enumeration  offers  no  difficulties.  The  minor  cycles  in  a 
given  DLri  organ,  on  the  other  hand,  are  merely  moving  loci,  at  which 
■  rtain  combinations  of  32  possible  stimuli  may  be  located.   Alterna- 
tively, looking  at  the  situation  at  the  output  end  of  the  DLA  organ, 
a  minor  cycle  is  a  sequence  of  32  periods  j  ,  this  sequence  being  con- 
sidered to  be  periodically  returning  after  every  1,024  periods  X   .  One 
might  say  that  a  minor  cycle  is  a  32  r  "hour"  of  a  1,022+  7    "day",  the 
"day"  thus  having  32  "hours'-1.   It  is  now  convenient  to  fix  one  of  this 
"hours",  i.e.  minor  cycles,  as  zero  or  ana  let  it  be  at  the 

same  time  at  the  outputs  of  all  256  DLA  organs  of  LI,  We  can  then 

attribute  each  "hour",  i.e.  minor  cycle,  its  number  0,  1,  ,  31,  by 

counting  from  there.  V.'e  assum?  accordingly  that  such  a  convention  is 
established —noting  that  the  minor  cycles  of  any  given  number  appear  at 

-89- 


r 


f 


t  \ 


the  same  time  at  the  outputs  of  all  256  DLA  organs  of  M. 

Thus  each  DLA  organ  has  now  a  number  jh  -   0,  1, ,  255 

(or  8-digit  binary),  and  each  minor  cycle  in  it  has  a  number  p  =  0,  1, 

,  31  (or  5-digit  binary).  _  A  minor  cycle  is-,  completely  defined  within 

M  by  specifying  both  numbers  i,  p.  .  Due  to  these  relationships  we  pro- 
pose to  call  a  DLA  organ  a  major  cycle* 

.  Fourth:  As  the  contents  of  a  miner  cycle  make  their 
transit  across  a  DLA  organ,  i.e.  a  major  -cycle,  the  minor  cycles  number 
p  clearly  remains  the  same.   When  it.  reaches  the  ''output  and  is  then 
cycled  back  into  the  .input  of  a  major  oycle  the  number. p  is  still  not 
changed  (since  it  will,  reach  the  output,  again  after  1,024  periods  T  , 
and  we  have  synchronism. in  all  DLA  organs,  and- a  1,024  f'   periodicity, 
cf.  above),  but  /u  changes  to  the  number  of  the  new  major  cycle.  For 
individual  cycling,  the  arrangement  of  Figure  19,  (a),  this  means  that 
/.<    ,  too,  remains  unchanged.  For  serial  cycling,  the  arrangement  of 
Figure  19,  (b),  this  means  that/a.  usually  increases  by  1,  except  that 
at  the  end  of  such  a  series  of,  say  s  major  cycles  it  decreases  by  s-1. 

These  observations .about  the  fate  of  a  minor  cycle  after 
it  has  appeared  at  the  output -of  .its -major  cycle  apply  as  such  when  that 
major  cycle  is  undisturbed,  i.e.  when  it  is  off  in  the  sense  of  13.2. 
When  it  is  on,  in  the  same  sense,,  but  in  the  first  case  of .13.3,  then 
our  observations  are  obviously  still  valid — i.e.  they  hold  as  long  as 
the  minor  cycle  is  not  being  cleared-.  When  it  is  being  cleared,  i.e. 
in  the  second  case  of  13. 3,  then  those  observations  apply  to  the  minor 
cycle  which  replaces  the  one  that  has  been  cleared. 


-90- 


( 


15.-0     The   code 
15.1  The   considerations   of   14.    provide   the   ba^is   for   a  fiamplete   classi- 

fication of  the    contents   of  K,    i.e.   they  enumerate   a  system  of  succe^ive 
disjunction  which   give  together   this   classification.      This   classification 
will  put   us    into   the   position  to  formulate   the   code  which  effects   the   logi- 
cal  control   of  CC,   and  hence    of  the   entire   device. 

Let   us   therefore   restate   the   pertinent  .definitions   and  disjunctions. 
The   contents    of  M  are. the   memory  units,   each  one  'being  character- 
ised by  the   presence    or  absence   of   a  stimulus.      It   can  b.e   used  to  repre- 
sent  accordingly  the   binary  digit   1   or  0-,   and  wo  will  at   any  rate   designate 
its   contont  by  the  binary  digit   i  =  1   or  0  to  which  it  corresponds   in  this 
manner,      (cf.    12.2. -and  12,5. ..with  76   )      Those   units   ar©-grouped   together 
zo  form  32«-unit  minor   cycles,   and  these  minor ,  cycles,  ar.o   the   ohtities 
>     which  will  acquire   direct   significance    in  the   code  which  wtf-will   introduce. 

*       i  •        • 

(cf.    12.2.)      Wb   denote   tho  binary  digits   which  make  up   tho   32  units    of  a 

minor   cycle,    in  their   natural  tomporal   scquonce,   by<i.  ,    i    ,.    i^ ,    i      . 

-  °        1        «J  31 

Th<.    minor   cycles   with  those   units   maybe  wrUton'  1=   (i    ,    i1  ,    i       ,    i_,    ) 

i     =  (i  >. 

v 

i.anor  cycles  fall  into  two  classes:   Standard  numbers  and  orders, 

(cf.  12.2,  and  14.1,)  Those  two-  ^Categories  should  be  distinguished  from 

each  other  by  their  respective  first  units  (cf.  12.2.)  i.e.  by  the  vulue 

of  iQ»  '.Vc  agree  accordingly,  that  i  =  0  is  to  designate  a  standard 

number  ,  and  i  =  1  an  order. 

o     ,.'.<■. 

15.2.    Tho  remaining  31  units  of  a  standard"  number  express  its 
binary  digits  and  its  3ign.   It  is  in  the  nature  of  al3.  arithmetical  opera- 
tion, specif ically  becauso  of  the  role  of .  carry  digits  ,  that  the  binary 

J 

-91- 


f 


' 


"^l      :!i~its   of  the   numbers   which  cn.cr    into   them,  must  be   fed    in  from  right   to 

left,    i.e.    those    with   the    lowest   positional    value.;    first.      (This    is   so 

because   the    digits   appear    in  a   temporal   succession  and  not   simultaneously, 

cf,    7.1.      The    details    arcs'- most   simply  evident    in  the    discussion    of  the 

adder    inT.3.)        The    sign  plays    the    role   of   the   digit   farthest    left,    i,.e. 

of   the   highest   positional  valuo   (cf.    8.1.)     Hence    it   comes    last,    i.e.    i      = 

0  designates   the   +   sign,   and    i_.    =  1   the  -   sign.      Finally  by  9.2     the 

binary  point   follows    immediately  after   the    sign  digit,    ana   the   number1? 

\   .        this   represented   must   '^r    moved  mod   2   into  the    interval  -1,   1.      That    is   - 

Jl  v  -   31 

=  hi  So  ^9  ---  h  gT-  \  2  <mod  2)«.  -*1  ;:a. 


5> 


15.3        Th     remaining  31   units   of  an  order,    on  the    othor   hand, 
must   oxpresr.   the    nature   of  this   order.      The    ordors   wore   classified   in  14..1 
into   four   classes    (a)    -    (d)  ,    and  those   were   subdivided   furthor  as    follows:. 
(a)    in   11.4,    (b)    in   14.2,    (b)    and   ^c)    in   14..3,.  14.4,   and  the   socond   re 
mark   in  14.5,      Accordingly,   the   following  complete   list    of   orders    obtains: 

(a)      Orders    for   CC  to   instruct  CA  to  carry  out   one    of   its   ten 
specific   operations   enumerated    in   11.4.      (This    is    (a)    in   14.1)      fro   desig- 
nate   these    operations   by  the.  numbers   0,   1,    2,.  — -,   9.,    in  the    order    in  which 
they  occur    in' 11.4,    and   thereby  place    ourselvc-    into  the    position  to   refer 
zo   any  one    of   then   by   it£    number  w  =   0,    1,    2,   _-_,  9,  which   is    best   reiven 
as   a  4^.digit  binary  (cf.,   hov/evor,  )      Rcgardm,:)  the*    origin  of 

the  numbers  which  entor  (a.",  variae  les)  into  those  operations  and  the  dis- 
posal of  the  result,  this  should  be  sai.;:  According  to  11.4,  the  formor 
come    from   ICA  and  o        and   the   latter   goes    to  0      ,   allin  CA  (cf.,    Figuros   16, 


-92- 


/' 


i 


( 


- 


17)      J_.    is   fed   through   I,,,,    ind   ln .    is   the   original    input   and  0       tho 
CA  GA  CA  or  q^ 

final    output   of   CA.      Consequently  thosj   aro  tho   actual   connecting  links 
between  U  ..      The   feeding  into   I- .    Will  be   described    in  (£),   (Y)  , 

(&)   below,  ooai   from  Op.will   bo   described   in  (</)  ,    (fc),   {.&)    below. 

Certain  oycr.tions  Qre  so  fast  (thay  can  be  handled  so  as  to  con- 
sume only  ■  ;.-  duration  of  a  minor  cycle),  that  it  is  worth  while  to  bypass 
;,-,>   when  disposing   of  tho  if   result,      (of.  ) 


-    provisions   for   cloarine   I       and  J  woro   described    in' 11.4. 

CA  C  » 

the    clearing  of  0       this    ought   to   bo   seid:      It  would   seem   natural 
CA 

\  .      . 

to  cle.r  Cq£  each   time    after    its   contents   have  been  transferred    into  h     (,cf. 

v) .        There   arc,   however,    cases,   when   it    is   preferable   n~t   to  transfer 

out   from  0      ,   .  nd    nd     to    :Icar   the   contents   of  Op.    Specifically:      In  tho 

f  the    operation  s    in   11.3    it   turned   out   to  bo   necessary  to 

hold    in  this   manner    in  Oq^     the  result   of  c,   previous    operation  -,      niter.- 

■ 

natively,    the   previous   operation  might   also  be   +,    i,    j,   or   oven   x,    cf.    tnoro, 
lOthor   instance:      If   a  multiplication  xy   is    carried   out,    with   an  0       which 

7M        contains,   say,    z   at   the    beginning   of  tho   operation,   then  actually   z   +  xy 

will    form    in  0        ;,cf.    tho   discussion  o.'   multiplication   in  7.7)      It  may 
.        0  A 

therefore   be   occasionally  desirable   to  hold  tho  rosult  of   an  operation,   which 

is   followed   .  y   a  multiplication,    in  0_ . .      Formation  of  sums    £_  xy  is   one 
nplc   of  this, 

rVb    ncod      ...  r:f   ro   an  additional   digit   c   =  0,  1  to  in  licatc-  whether 

0        should   or    should    no*,   be    clcarod  after   the    operation.      'Arc    lot   c   =  0  ox- 
press    tho    former  t    and   c=    1   tho . latter. 


-93- 


c 


£ 


(3)    Orders  for  X'  to  cause  the  transfer  of  a  standard  number 

3) 

from  a  definite  miner   cycle    in  M  to  CA.    (This    is    (b)    in  14.1,   type    (b'' )    of 

14.?)      iho  minor   cycle    is   dcfinod,by  the   two   indices   u,    p     (cf.    the    third 

remark   in   14.5;      ?ho   transfer    into  CA   is,   more   precisely,    one-  into   I        (c;'. 

CA 

(a)  above). 

(Y)   Orders  for  CC  to  cause  the  transfer  of  a  standard  number 
which  follo-.vs  immed iately  upon  the  order,  into  CA,   (This  is  the  immediate 
transfer  of  the  second  remurk  in  14.5  in  the  variant  which  corresponds  to 

1  above.)   It  is  simplest  to  consider  a  minor  cycle  containing  a  standard 
number  (the  kind  ;.  nalyzed  in  15,2)  as  such  an  order  per  se,   (This  modifies 

at 

cttiterncnt    lpc.    cit,    somewhat:      The    standard   number    in  question    is   -^s*    i< 

the   minor   cycle   following   immediately  upon   a  minor   cycle   which  has    just 
given  an   order   to   CC ,    then  the    number  will    automatically  operate    as    an 
immediate   transfer   order    of  the   typo   described.      (cf.    also  the    pertinent 
remur/.s    in  (£)    and    in   (^  )    bolow.)      The   transfer    into  CA   is   again   or.e    into 

*CA  ^cff    ^ a^    or    ^'J^    ~bovo.) 

(</)   Orders  for  CC  to  cause  the  transfor  of  a  standard  number  from 

m 

CA  tc  a  definite  minor  cycle  in  M,   (this  is  (jb)  in  14.1,  type  (bM)  in 
14.2)   The  minor  cycle  in  M  is  defined  cy  tho  two  indices  u,  p,  as  in  (^) 
above.   Tho  transfer  from  CA  is,  more  precisely,  one  from  0n     —  this  was 
discussed,  together  with  the  necessary  explanations  and  qualif ioo tions"}  in 
(a)  above. 

(£.)   Orders  for  CC  to  cause  tho  transfer  of  a  standard  number 
from  CA  into  tho  minor  cycle  which  follows  immediately  upon  the  one  contain- 
ing this  ordor.   (This  is  the  immediate  transfer  of  the  second  remark  in 

I 

-94- 


1/ 


e 


14.5,  in  the  variant  which  corresponds  to  {(f)    abovo.)   The- transfer  from 


CA  is  again  ono  from  0   (cf.  (a)  or  (</)  above.) 

In  this  case  the  CC   connection  passes  from  this  transfer  order  on 
to  the  next  minor  cycle,  into  which  the  standard  number  in  question  is  just 

being  sent.   There  would  bo  no  point  in  CC  now  obeying  (y),  and  £ 
this  number  sack  into  CA  —  also,  there  might  bo  timing  difficulties.   It 
is  best,  therefor.,  to  except  this  case  explicitly  from  the  operatic,  of 
(y).  I.e.:   (Y)  is  invalid  if  it  follows  immediately  upon  an  (£).  j 

(6)  Orders  for  CC  to  cause  the  transfer  of  a  standard  number 

from  CA  into  CA.   (This  is  an  operation  of  CA,  the  usefulness  of  whici  v 
recognized  in  11.2  cf.  also'  )   More  precisely,  from  0c;  Lntc 

(cf.  (a)  above) 

{i)  Orders    for   CC   to  transfer    its    own  connection   with  U  to 

)     a  definite   minor   cycle    (elsewhere),    in  M.      (This    is   (c)    in   14*1)      The 

minor    cycle    is   M    is   defined   by  the    two    indices    u,    p,    as    in   (£ )     •'- 

Note,    that'  a  (|3)   could  boropla  ed  by  a  {I),   considering  (y).      tft'Ly+QQ^) 

s  except 


The   only  difference    is,   that   (/)    is   a   permanent  transfer,   while    (0)    is   a 
transient   one.      This   nay  servo   to   place   additional   emphasis    on   the    corres- 
ponding considerations    of    14.3   and    14.4. 

(y\)  Orders    controlling  the    operation   of  the    inpat   and   the    out- 

put   of   the   device    (i.e.    I   of   2.7   and   0  of   2.8    )    (This    is    (d)    in   14.1) 
As    indicated    in   14.1,   the   discussion   of   those    orders    is   bottom   delayed 
(cf.  ). 


-95- 


J\ 


c 


~> 


15.4        Let    us   nov,   convDarc    the    nunrt    rs    of    digits   necessary  to 

express   th  orders   with  tho   number   of  av      1       1        i(  its    in  ..  minor   cycle  _ 

31,   as    stated   at    tho   beginning  of   15.3. 

Io  begin   .vith  wc   have    in   (t^)    -   (>0   3   typos    of   orders,   to 

distinguish  those    from  each   other  re   uirjs   3   digits.      ttext,    tho    types    (a)   _ 

(£)    (    wo    postpone    (*))  ,    cf.    above)    have    those   requirements:    (,a)     .    s1    .: ;  :C:ify 

^AoljJt   or    char    Oca 
the   number   w,    i.e.    4    digits,    plus    tho    digit   c   -   all    together   5   dibits. 

(0),   as  -.veil   as    (d)    and   (£),   must   specify  the   numbers   yv  and   p  ,    i.e.   8   + 
5  =  13  digit's,    (y)    is   outside   this   ca/cegory.    (£)»   as  well  as   (.6),   requires 
no  further   specif icutions-; 

"cither   o:    thssc   uses   thc31   available   dibits   very  efficiently. 
Consequently  we  might    consider    putting   several   such    order.:    into    one   minor 
cycle.      On  the    other   hand   such  a   tendency  to  pool    orders    should  bo   kept 
|      ithin  very  d^inite    limits,   for   the   following  reasons. 

First,    pooling  several   orders    into   one  minor   cycle   should  be  . 
avoided,    if    it  requires   tho   simultaneous   performance   of   several   operations 
(i.e.    violates    the    principle    of  5.6.)      Secdd,    it   should   also  be   avoided 
if   it   upsets    the   timing   of  the    operations.      Third,    the   entire   matter    is 
usually  net  important   from  the   point   of  view   of  the    total  memory  capacity: 
Indeed;   it  reduces    the    number    cf  those   minor   cycles    only,    which   are    used 
for  "logical    instructions,    i.e.    for -the    purpose    (b)    in  2.1,    and   these 
represent    usually   only  a   small   fraction   of    the   total   capacity    of  M  (cf. 
(b)    in  12.3   and  ).      H-ncc  tho   pooling  of   orders    should   rather  be 

carried   out   from  tho   point    of   view  of   simplifying  the    Logical   structure   of 
tho    code.  •     •  ' .     , 


) 

-96- 


( 


T) 


...O.r.        Mioso   considerations   discourage   pooling  several   orders   of 
the       -re    (a)   -  besides   this   would   often  not   be    logically   possible    either, 

without    intervening   orders    of   the   types  •,(£)   -   (5).      Combining  two   orders 

of  the    typos    (|3),    (<7 ) ,    (£)    is    also  dubious    from   the   above    points    of     'ic  f, 

bosidos    it    would    leo  •:■    only  31-3-13-13  =  2  digits    free,    and  this    (although 

it  could   be    incrc  .sod    by  various   tricks   to   o)    is   uncomfortably   low:      It 

is    advisable    to   conserve    some    spare    capacity   in  the    logical   part   of   the    co;c 

(i.'e.    in  tho    orders),    since   later- on  changes   might  bo   dosirablo.      (S.g. 

it  May  bocomo   advisablo    to    increase   the   capacity   of  LI,    i.e.    tho   number 

:    of  m  jor   cycles,    i.e.    the    number  Q   of  digits    oT  u.      For   an   other   reas    n 
of. 

The   boat   chance   lies    inno-olinf    an   operation   order   (a)   with  cr- 

•£  controllin  the  transfer  of  its  variables  into  Ca  or  the  transfer  of 
its  result  out  of  CA.  Both  .types  may  involvv  13  d  ig  its  orders  (namely  (J3) 
or   (a)    ),   henc<_    are    c  ..ount   on  polling   (ot)    with  more    than   one   such 

lor    (cf.    ti.      ai  ..vo   estimate    plus   the   5   digits    required  by  (a);   !),      Now 
.  1     u:   i    11^    requires    sra.isforr ing  two   variables    into   CA,    honC'~    tho 

yst  ,     .1   procedure    consists    in   pooling    {'a)    with  the   disposal 

of    i  .ilt.      I.e.    (a)    with   (cf)    or    (£)    or    (6).      It   should   be    noted  that 

every   (<?)  ,    ( &) ,    ( &)  ,  -i,e.    transfer   frorr.  CA  must   be   preceded   by  an   («,), 
and    every  (P),    (Y),-i,o;    transfer    into  CA,    just  be    followed  by  an   (a). 
Indeed,    those    transfers . are    always    connected    with  an   (a)    operation,    the 
only   possible    exception  would  be    an  U  to  I',  transfer,    rtoutcd   through   (a), 

-   even   this    involves '•  an   (a)    operation   (i   or  j    in   11.4,    cf.    thoro   and 
11.2   ).      Cc  Fitly  c   dors    (c1),    (O    (a)   will    always    occur   pooled  with(  a)  , 

and     >rdora    (j3),    (Y)    will       1;    ys    occur    .'-lone.       ( a) ,    too,. may  occasionally 

-97- 


c 


( 


1 


3 


occur    clone:       If   the    result    of   th>  ■  .    (a)    is    tc 

(cf.     cl  ---rt    oh    (a)     in    15.3)-,    r.      •      -    '..ill    ...        LI;     not 

-  .,      .-■■      ;  •        Lisposc    of    this    rc3ul  .y 

also    (cf.    th  -      .cit.)       Wc    shall       ;ep  be  ssib-ili    Los    .    sn: 

th  «  not'  bi  litioi  1   of   th      result,      nd  :ho 

ca.se    '  -  j  pooled    .vitb  1   order.      Orders    (s) 

urc   of   a   sufi    cic    lly    sxct         onal    logical   ch  ractor,    r       justify  that    they 

Lone, 

-    i  disra   ard   (y),   which  "is    in   roality  c   st 

'■or   -   the  7   folic  types    ::'  oriors:    (a)+(V),    (  a) +(.6)  ,-  (  a)  ,    (@)  ,    (f )  , 

X  A  J 

|f).      They  re  ,l3=lfft   S,o,r,    15,    13   di;ixs    (    _■     liaucgard    (*£)  , 

-1    be   discus     3d    1    tcr)    plus    3   digits    to   distinguish  the    typos    from  c 

r,    pluJ    one    ii    it    (i  pi)    to  express   that   an   crier    is    involved. 

i 
^hc    totals    ere    22,    9,    9V  9;i    17,    17      digits.      This    is    an    average    jffic       icy 

)      of   -   50%    in      :  il  -    the   32  digits    of    Lor   cycle.       rhis    offecienev 

pan  he    considered   adequate,    in  view   of  -the    third   rein,  rk   of   15.4,    and    it 
it    the    same   ti    -        : . vf crtablc    spare    c.  pacity  (cf.    the    beginni  i( 
of   1    .    )'. 

15.6        r'c    arc    now   in  the    position    to   formulate    our   cede.      This 
formulati'  n  will    t  -    srooented    in  tho* following  manner: 

r    ■  t      characterize   all   possible  miner   cycles 

vice.      Thoac   are   standard   numbers   and    order, 
•      :  ;     '.      cri      i    in   15.1  -   1?,5.      In^  feho    toblo    which 
fol]  ■■:        pecify  the    f  ur -following  things    for   c    c     possible  minor 

cycle:    (*:)    The    type    ,    i.e.    its   relationship  to  th..   cLissif ication   (a)    -    (>•)) 


-98- 


( 


< 


f 


k 


c 


of   15.3,    :nd    to  the    pooling   procedures    of   15.5.      (II)      The   moaning,    as 
"^  in    l; .  1   -   15.5,      (III)      The   short,  symbol,    to  be    used    in   verbal 

or  written   iiscussidns    of   rhc   code,   and    in  particular    in  all    further 
.     lyss;    ...'    thia   papor,    and   when  sotting  up   pre    lcir.s    for  the    device. 
(Cf.  )      (IV)    The    zodo   symbol,    i.e.    the   32  binary  digits 

i}i    io, ,    I31,   which  correspond  to  the   32  units    of  the   minor    cyo3o    i:. 

1 

tic    over,    there  will   only  bo   partial   stctc:r.ontS    on  thin    I 
.     int    at    this    time_,    the   precise   description  will  be   given   lator(cf 

the    numbers   (binary  integers)   which  occur    in    ;he£ 
\  jcIc,   wc    obsorvi     this:      Those   numbors    arc   u,   p.   w,    c,    .  Vc    . .  ill    3    n  tc 

its    (in   the    usual,    left  tc   right,    order)    by  u        ,    u    j 

ft »#«*.   1  .;  w3, ,   wo5   c. 


I 


-99- 


c 


Table, 


-ji: 


en) 


-    ..i:\rd 

er 
or 
der 
. 


for    bhe     .  fif.be  r 

:  l 


■i  •        i       


i  V/2T 


defined   b; 

1-v      i 

(.mo      : 


■'j\ 
1  -    i=%- 


i_,       ;   the    sign:    0   for   +,    i   for  -. 

If  CC    i^    :   >nnected   to   this   minor   cycle,    then   it 
i     ■•  >rder  .causing   the    tr   :-.:  -    r     •:' 


oO    +(f) 


into    ICf 


.'.       Ipes   no      ar  ily   however    Lf    t]  is 


oi    eye]      ,'   ]     iv/s  •      an   orier 

W  -:.  A     jr     .:..    .       . 


r  i'" . "    t o   c  fc   the    operation   ;■    in  CA  and  to 

of    t;       result,      w    is    fro...  the    list   of 
11. 4.    T\  re    the    operations    of    11.4,    .;itr. 

ir   current   numbers    .;   and    their    symbols    ;r. 


. .-  ler 
(*)    +(©) 

~) 


w  ->up 

or 
*Jli  — >up 


(a) 


til;   :  the    result    is    to   be   held    in   Oca. 

— >up  means,  • ..  ,t   the   result    is    to  be   trans-           i 

fen     i    intc  :  \  \  minor   cycle  pi.;   the   major   cycle 

u;   — jf,    th:i  U     L:           be   transferred    into   the 

:;/:.,.  :    tely   following   upon   the    order 

— ?  .,    that    .  is    to  be   transferred    into    I      ; 

:C  ->,    that    .  .  .          .'  -     L:     n   nted    (apart   :ror..   h). 


wh 


)rder 
<0 


& 


;r    to   '  >r   tho   number    in  the   minor 

.-  p  .  yjor    cycle   u    intc    Ic    . 


o    connect   CO   ,vith  the   minor   cycle p  in 
jor   cycle  u. 


(  IV) 


Code 


A«~*  ut 


C<f— up 


h  =  l 


-100- 


( 


( 


1 


Remark:  Orders  w  (or  wh) — »up  (or  f)  transfer  a 
standard  number  k.      from  Cn   into  a  miner  cycle.  If  this  miner  cyclv: 
is  of  the  type  N  \   (i.e.   iQ  -  0) ,   then  it  should  clear  its  31  ii 
representing  \     ,   and  accept  the  31  digits  of  t,  .  If  it  is  a  riinnr 
cycle  ending  in  up  (i.e.  i0  =  1,  order' w  — ?-up  or  wh  — %  up  or  k  <--  v.o   :' 
C  <—   up),  then  it  should  clear  only  its  13  digits  represent inr  up,  ■'■!.. 
accept  the  last  13  digits  of  \  I 


i 


v 


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Hat": 


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